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H i C N Households in Conflict Network The Institute of Development Studies - at the University of Sussex - Falmer - Brighton - BN1 9RE
www.hicn.org
Networks in Conflict: Theory and Evidence from the
Great War of Africa1
Michael D. König2, Dominic Rohner3, Mathias Thoenig4 and Fabrizio Zilibotti5
HiCN Working Paper 195
February 2015
Abstract: We study from both a theoretical and an empirical perspective how a network of military
alliances and enmities affects the intensity of a conflict. The model combines elements from
network theory and from the politico-economic theory of conflict. We postulate a Tullock contest
success function augmented by an externality: each group’s strength is increased by the fighting
effort of its allies, and weakened by the fighting effort of its rivals. We obtain a closed form
characterization of the Nash equilibrium of the fighting game, and of how the network structure
affects individual and total fighting efforts. We then perform an empirical analysis using data on
the Second Congo War, a conflict that involves many groups in a complex network of informal
alliances and rivalries. We estimate the fighting externalities, and use these to infer the extent to
which the conflict intensity can be reduced through (i) removing individual groups involved in the
conflict; (ii) pacification policies aimed at alleviating animosity among groups.
1We would like to thank Andr´e Python, Sebastian Ottinger and Nathan Zorzi for excellent research assistance. We are grateful for helpful comments from Daron Acemoglu, Alberto Alesina, Ernesto Dal Bo, Alessandra Casella, Melissa Dell, David Hemous, Macartan Humphreys, Massimo Morelli, Benjamin Olken, Maria Saez Marti, Uwe Sunde, Fernando Vega-Redondo, David Yanagizawa-Drott, Giulio Zanella, and conference and seminar participants of the ESEM-Asian Meeting (Taipei, 2014), IEA World Congress (Jordan, 2014), NBER SI 2014, and at the universities of Bocconi, Bologna, CERGE-EI, Columbia, Harvard, MIT, European University Institute, IMT, INSEAD, IRES, Lausanne, LSE, Luxembourg, Marseille-Aix, Munich, Nottingham, Pompeu Fabra, SAET, Southampton, St. Gallen, Toulouse, ULB and Zurich. Mathias Thoenig acknowledges financial support from the ERC Starting Grant GRIEVANCES-313327. 2Department of Economics, University of Zurich. Email: [email protected]. 3Department of Economics, University of Lausanne. Email: [email protected]. Dominic Rohner gratefully acknowledges funding from the Swiss National Fund grant 100017 150159 on “Ethnic Conflict”. 4Department of Economics, University of Lausanne. Email: [email protected] 5Department of Economics, University of Zurich. Email: [email protected].
1 Introduction
Alliances and enmities among armed actors – be they rooted in history or in mere tactical consid-erations – are part and parcel of warfare.1 In many episodes, especially in civil conflicts, they areshallow links that are not sanctioned by formal treaties or war declarations. Even allied groupsretain separate agendas and pursue self-interested goals in competition with each other. The com-mand of armed forces remains decentralized, and coordination is minimal.
Understanding the role of informal networks is important, not only for predicting outcomes,but also for implementing policies to contain or put an end to violence. These may be diplomaticinitiatives promoted by international organizations to restore dialogue and reduce animosity be-tween conflict participants, or military interventions of external forces against specific groups. Yet,with only few exceptions, the existing political and economic theories restrict attention to conflictsamong a small number of players, and do not consider network aspects. In this paper we constructa theory of conflict focusing explicitly on informal networks of alliances and enmities, and apply iteconometrically to the study of the Second Congo War (1998-2003) and its aftermath.
The theoretical benchmark is a contest success function, henceforth CSF (see, e.g., Grossmanand Kim 1995, Hirshleifer 1989, Skaperdas 1992). In a standard CSF, the share of the prize accruingto each group is determined by the amount of resources (fighting effort) that each of them commitsto the conflict. In our model, the network of alliances and enmities modifies the sharing rule of astandard CSF by introducing additional externalities. More precisely, we assume that the shareof the prize accruing to group i is determined by the group’s relative strength, which we labeloperational performance, henceforth OP. In turn, the OP is determined by group i’s own fightingeffort and by the fighting effort of its allied and enemy groups. The fighting effort of group i’sallies increases group i’s OP, whereas the fighting effort of its enemies decreases it. Thus, eachgroup’s fighting effort affects positively its allies’ OP and negatively its enemies’. Instead, the costsof fighting are borne individually by each group. This raises a motive for strategic behavior amongboth enemy and allied groups. In particular, there is not even coordination between allies: allagents determine their effort in a non-cooperative way, and alliances are loose links.2 The complexexternality web affects the optimal fighting effort of all groups.
We provide an analytical solution for the Nash equilibrium of the game. Absent other sourcesof heterogeneity, the fighting effort of each agent hinges on a measure of network centrality which isrelated to the Bonacich centrality (Ballester, Calvo-Armengol and Zenou 2006). Our centrality isapproximately equal to the sum of the Bonacich centrality related to the network of enmities, andthe (negative-parameter) Bonacich centrality related to the network of alliances. The equilibriumshare of the prize accruing to each player and the associated welfare (i.e., the share of the prizenet of the fighting effort) have simple expressions. Intuitively, a group’s welfare is increasing in thenumber of its allies and decreasing in the number of its enemies.
The ultimate goal of the theoretical analysis is to predict how the network of military alliancesand rivalries affects the overall conflict intensity. This is measured by the sum of the fighting efforts
1Ghez (2011) distinguishes between tactical, historical and natural alliances. Tactical alliances are formed ”tocounter an immediate threat or adversary that has the potential to challenge a state’s most vital interests” (p.20). They are instrumental and often opportunistic in nature. Historical alliances are more resilient insofar as theyhinge on a historical tradition of cooperation. However, they often remain informal. Natural alliances imply a moreprofound shared political culture and vision of the world (e.g., Western Europe and the U.S.). Contrary to tactical andhistorical alliances, natural alliances often are formalized relationships. Our study focuses on tactical and historicalalliances/enmities. In our theory, natural allies can be viewed as merged actors acting in a perfectly coordinatedfashion.
2In some historical examples alliances are more than shallow links. Our theory can incorporate strong alliances(e.g., between the U.K and the U.S. during WWII) by treating them as merged unitary groups.
1
of all contenders (total rent dissipation), which is our measure of the welfare loss associated witha conflict. We show that network externalities are a key driver of the escalation or containment ofviolence.
As an illustrative example, we analyze a ”regular” network, in which the number of alliancesand enmities is invariant across groups. We show that conflict intensity and rent dissipation aremaximized when all groups are connected by enmity links. In this case, the outcome is a Hobbesianpre-contractual homo homini lupus society. When externalities are sufficiently strong, the cost ofconflict may offset the social surplus in this society. To the opposite extreme, conflict and rentdissipation are minimized (and, possibly, vanish) in networks where all groups are allied, as inRousseau’s well-ordered society governed by the social contract. The well-ordered society may bea surprising outcome in a non-cooperative contest between self-interested agents. The crux of theresult is that the marginal product of fighting effort decreases in the number of alliances, becausethey dilute the marginal benefit from exercising individual fighting effort. For sufficiently strongalliance externalities, the incentive to fight vanishes altogether. The standard free-riding problemhas a benign effect in our model, since war effort has no social value. The peaceful outcome can beviewed as representing a society in which the system of institutional checks and balances reducesthe return to opportunistic behavior.
In the second part of the paper, we perform an empirical analysis based on the structural equa-tions of the model. We focus on the Second Congo War, sometimes referred to as the ”Great AfricanWar”. This is a big conflict, with an estimated death toll of 3-to-5 million lives (Autesserre 2008;Olsson and Fors 2004). It involves many groups, and a rich network of alliances and enmities.3 Toidentify the network of alliances and enmities we use information from the Stockholm InternationalPeace Research Institute (SIPRI), supplemented by information from the Armed Conflict Loca-tion Event Database (ACLED). We assume that the network is exogenous and time-invariant – anassumption that conforms with the data for the period considered (see the discussion in section3.3) – and proxy fighting effort by the number of fighting events in which each group is involved.The estimated networks of enmities and alliances feature numerous intransitivities, showing thatthe Second Congo War can by no means be described as a fight between two unitary opposingcamps (see Figure 6 below). Our estimation strategy exploits panel variations in the yearly fightingefforts exerted by 85 armed groups over the 1998-2010 period. Controlling for group fixed effectsand time-varying unobserved heterogeneity we regress the individual fighting effort of each groupon the total fighting efforts of its degree-one allies and enemies, respectively. Since these are en-dogenous and subject to a reflection problem which is standard in regressions involving networks(see Bramoulle, Djebbari and Fortin 2009; Liu et al. 2011), we use an IV strategy similar to thatused by Acemoglu, Garcia-Jimeno and Robinson (2014). In particular, our identification exploitsthe exogenous variation in the average weather conditions facing, respectively, the set of alliesand of enemies of each group. Our focus on weather shocks is motivated by the recent literaturedocumenting that these have important effects on fighting intensity (see Dell 2012, Hidalgo et al.2010, Jia 2014, Miguel, Satyanath and Sergenti 2004, and Vanden Eynde 2011). Without imposingany restriction on the estimation procedure, we find that the two estimated externalities have the(opposite) sign pattern, which aligns with the predictions of the theory.
After estimating the size of the network externalities, we perform two sets of policy-orientedcounterfactual experiments. First, we remove sequentially each of the groups in conflict while lettingall surviving group re-optimize their fighting effort. This key-player analysis is a policy-relevantexercise, as it can help international authorities to single out armed groups whose decommissioningwould be most effective for scaling down a conflict. Interestingly, we find that while on average
3More details about the historical context of this conflict are provided in Section 3.1.
2
more active groups have a higher rank in the key player analysis, the relationship is far fromone-to-one. For instance, the two factions of the Rally for Congolese Democracy (RCD) – Gomaand Kisangani factions – rank, respectively, first and third in terms of their involvement in warepisodes. Yet, their hypothetical removal would not contribute much to scaling down the conflict,because of their position in the network. According to our estimates, removing RCD-K wouldactually increase total violence, as the muted fighting of this group would be more than offset bythe average increase in the effort of other armed groups. In contrast, the removal of the Huturebel group denominated Democratic Forces for the Liberation of Rwanda (FDLR) would yield a14% reduction in violence. Our analysis also highlights the prominent role of foreign actors: eightout of the twenty groups bearing the largest responsibility for the escalation of violence are foreignnational armies. Removing all foreign troops would reduce total violence by 24%, a large effect.
We also study a policy aimed at pacifying armed groups without removal, i.e., rewiring enmitylinks into neutral or alliance relationships. First, we show that pacifying all enmities in the Demo-cratic Republic of Congo (henceforth, DRC) would yield a reduction in fighting by between 54%and 91%. Next, we show that a more realistic policy aiming to pacify individual groups could alsobe effective. For instance, rewiring all enmities of the FDLR into neutral relationships would yielda de-escalation of the conflict by 13%.
Our contribution is related to various strands of the existing literature. First, our paper islinked to the growing literature on the economics of networks (e.g., Acemoglu and Ozdaglar 2011,Bramoulle, Kranton, and d’Amours 2014, Jackson 2008, Jackson and Zenou 2014). There exist onlyvery few papers in the literature studying strategic interactions of multiple agents in networks ofconflict. Franke and Ozturk (2009) study agents being embedded in a network of bilateral conflicts,where agents can choose their fighting efforts to attack their neighbors. However, they do not allowfor alliances. Moreover, they only characterize equilibrium efforts for particular networks (regular,star-shaped, and complete bipartite graphs), while we provide an equilibrium characterization forany network structure. Huremovic (2014) extends their model introducing endogenous fightingefforts in contests. His model differs from ours in several respects: there are no alliances, and theconflict is restricted to a collection of pairwise contests (while in our model many agents competeover a common prize). Neither of these studies tests the model empirically.
Two recent theoretical papers study the endogenous formation of networks in models of conflict.Hiller (2012) considers a model where agents can form alliances to coerce payoffs from enemies withfewer friends. Jackson and Nei (2014) study the formation of alliances in multilateral interstatewars and the implications on trade relationships between them, showing that trade can have amitigating effect on conflicts. Neither of these papers consider, as we do, an endogenous choice offighting effort. Different from these papers, we do not analyze the determinants of the network,which we infer from the data and treat as exogenous. Extending the analysis of endogenous networkformation to large-scale networks including both enmities and alliances like the one in the CongoWar is a challenging endeavor that is beyond the scope of this research.4
All of the above papers provide theoretical contributions. Our paper provides an estimation ofthe model based on the structural equations of the theory, and uses the estimates of the structuralparameters to perform a key player analysis. In this sense, our paper is related to recent work byAcemoglu, Garcia-Jimeno and Robinson (2014) who estimate a political economy model of public
4From an empirical standpoint, most group characteristics that determine whether each two groups are friends,enemies or neutral are unobservable to the econometrician. The challenge is to predict how such shocks would reshufflenetwork links. We conjecture that the endogenous reshuffling of alliances and enmities in response to weather shocks– which is the source of identification in our model – must be rare. Consistent with this conjecture, we see only verylittle reshuffling of alliances during the Second Congo War. Thus, we do not view network endogeneity as a majorthreat to our econometric strategy.
3
goods provision using a network of Colombian municipalities. Their empirical strategy is related toours, although they use historical variations in players’ characteristics while we use panel variations(exogenous shocks in rainfall).
The key player analysis in strategic games with networks is pioneered by Ballester, Calvo-Armengol and Zenou (2006). The authors determine equilibrium effort choices in a game of strategiccomplements between neighboring nodes, and identify key players, i.e. the agents whose removalreduces equilibrium aggregate effort the most. However, their payoff structure is substantiallydifferent from ours, and does not incorporate an environment in which agents compete for commonresources. More recently, the key player policy has been studied empirically. Liu et al. (2011) testthe key player policy for juvenile crime in the United States, while Lindquist and Zenou (2013)identify key players for co-offending networks in Sweden.
Further, our study is broadly related to the growing politico-economic conflict literature. Pa-pers in this literature typically focus on settings with two large groups facing each other, withoutconsidering network relationships. A few papers consider multiple groups comprising each a largenumber of players, and study collective action problems. Esteban and Ray (2001) show that theOlson paradox does not generally hold and that sometimes large groups can be more effective thansmall groups. To this purpose they build a model with n different groups composed each of avarying number of individual players. Their model is different and complementary to ours: theyhave no network structure, but focus on a setting comprising n different cliques, where there areno links between cliques. Related to that, Rohner (2011) constructs a conflict model allowing forn aggregate ethnic groups comprising a given number of individual players.
Our paper also shares some commonalities with the literature on the role of alliances in settingswith either three players, or with identical players (see the surveys in Bloch 2012, and Konrad2009 and 2011). Some papers note that alliances may be socially desirable since they reduce rentdissipation in wars (e.g. Olson and Zeckhauser 1966). Other papers regard the mere existence ofalliances as a ”puzzle” given the free-riding (collective action) problem they generate, and the factthat victorious coalitions must share resources when victorious (see Esteban and Sakovics 2004,Konrad and Kovenock 2009, and Nitzan 1991).
Finally, our paper is related to the empirical literature on civil war, and in particular to therecent literature that studies conflict using very disaggregated micro-data on geo-localised fightingevents, such as for example Dube and Vargas (2013), Cassar, Grosjean, and Whitt (2013), LaFerrara and Harari (2012), Michalopoulos and Papaioannou (2013), Rohner, Thoenig, and Zilibotti(2013b). In a recent interesting paper on the conflict in the DRC, Sanchez de la Sierra (2014)studies how price shocks of particular metals (cobalt, gold) affect the incentives of armed groupsto establish control of resource-producing villages in Eastern Congo.
The paper is organized as follows: Section 2 presents the theoretical model and characterizes theequilibrium; Section 3 discusses the application to the Second Congo War and presents the mainestimation results. Section 4 performs a number of robustness checks, while Section 5 performssome policy counterfactual analyses (key player and pacification policies). Section 6 concludes. Anappendix contains some technical material.
2 Theory
2.1 Environment
We consider a population of n ∈ N agents (armed groups) whose interactions are captured by anetwork G ∈ Gn, where Gn denotes the class of graphs on n nodes (vertices). Each pair of agentscan be in one of three states: alliance, enmity, or neutrality. We represent the set of bilateral states
4
by the matrix A = (aij)1≤i,j≤n where aij ∈ {−1, 0, 1}. More formally, A is the signed adjacencymatrix associated with the network G, where, for all i 6= j,
aij =
1, if i and j are allies,
−1, if i and j are enemies,
0, if i and j are in a neutral relationship.
Note that a neutral relationship is modeled as the absence of links.Let a+ij ≡ max {aij , 0} and a−ij ≡ −min {aij , 0} denote the positive and negative parts of aij ,
respectively. Then, aij = a+ij − a−ij , respectively, for all 1 ≤ i, j ≤ n. Similarly, A = A+ − A−
where A+ = (a+ij)1≤i,j≤n and A− = (a−ij)1≤i,j≤n. We denote the corresponding subgraphs as G+
and G−, respectively, so that G can be written as the graph join G = G+ ⊕G−. Finally, we defineby d+i ≡
∑nj=1 a
+ij and d−i ≡
∑nj=1 a
−ij, respectively, the number of allies and enemies of agent i.
The n agents compete for a prize whose total value is denoted by V > 0. We assume agents’payoffs to be determined by a generalized Tullock contest success function (CSF) (cf. Tullock 1980,and Skaperdas 1996). The CSF maps the relative fighting intensity each agent devotes to a conflictinto the share of the prize he appropriates after the conflict. More formally, we postulate a payofffunction πi : G
n × Rn → R given by
πi (G,x) = ζ i + Vϕi (G,x)∑nj=1 ϕj (G,x)
− xi (1)
where ζ i ≥ 0 is an exogenous endowment, x ∈ S is a vector describing the fighting effort of eachagent, and ϕi ∈ R is agent i’s operational performance (OP). The latter is assumed to depend onagent i’s own fighting effort, as well as on his allies’ and enemies’ efforts. More formally, we assumethat
ϕi(G,x) ≡ xi + βn∑
j=1
a+ijxj − γn∑
j=1
a−ijxj , (2)
where β, γ ∈ [0, 1] are spillover parameters from allies’ and enemies’ fighting efforts, respectively.Note that the specification of equation (2) implies no heterogeneity across agents other than theirposition (i.e., the number of allies and enemies) in the network. We introduce other sources ofheterogeneity (e.g., military power) in Section 2.6 below.
Equation (2) postulates that each agent’s OP increases in the total effort exerted by its allies anddecreases in the total effort exerted by its enemies. These externalities compound with those alreadyembedded in standard CSFs, which equation (2) nests as a particular case when a+ij = a−ij = 0 for all
i and j. In this case, πi(G,x) =(xi/∑n
j=1 xj
)V − xi, and each agent’s effort imposes a negative
externality on the other agents in the contest only by increasing the denominator of the CSF.Consider, for example, a network such that a+kk′ = 1 and β > 0, for one and only one pair
of agents (k, k′) (while a−ij = 0 for all i, j = 1, . . . , n). Then, πk(G,x) = V (xk + βxk′) /(∑n
i=1 xi+ β(xk + xk′)) − xk. In this case, an increase in the effort of k′ affects the payoff of k via twochannels: (i) the standard negative externality working through the denominator; (ii) the positiveexternality working through the numerator. Thus, holding efforts constant, an alliance betweentwo agents increases the share of V jointly accruing to them, at the expenses of the remaininggroups. To the opposite, enmity links strengthen the negative externality of the standard CSF.
Note that the OP could in principle be negative (for instance, when an agent has many enemiesexerting high effort). We interpret a negative OP as a situation in which a group is expropriatedof part of its endowment (ζ i) – since utility is linear in income, this entails no loss of generality.
5
In the Nash equilibrium characterized in Section 2.2 below, however, all groups’ OP is positive.5
Note further that there is no ”participation decision”, namely, belligerent groups cannot decide toabstain from the conflict and live from their endowment. This has a natural interpretation: groupscannot flee from the country, and not making any fighting effort would expose them to other armedgroups’ looting and ransacking activity. For technical reasons, we impose a lower bound to thechoice set, i.e., x ∈ S = [x1,∞)× [x2,∞)× ...[xn,∞).
2.2 Equilibrium Fighting Effort
In this section, we characterize the Nash equilibrium of the contest. More formally, each agentchooses effort (xi) non-cooperatively so as to maximize πi (G, [xi,x−i]), given x−i. The equilibriumis a fixed point of the effort vector.
Using equations (1)-(2) yields the following system of first order conditions (FOC), for i =1, 2, . . . , n:
∂πi (G,x)
∂xi= 0 ⇐⇒ ϕi =
1
1 + βd+i − γd−i
1−
1
V
n∑
j=1
ϕj
n∑
j=1
ϕj.
We assume that, for all i:βd+i − γd−i > −1. (3)
This condition is both necessary and sufficient for the second order condition to hold for all playersat the equilibrium (see Proposition 1 and its proof in Appendix A). In the empirical analysis, wecheck that this condition holds for our estimates of β and γ.
Rearranging terms allows us to obtain a simple expression for the equilibrium OP level,
ϕ∗i (G) = Λβ,γ (G)
(1− Λβ,γ (G)
)Γβ,γi (G) × V, (4)
and for the share of the prize accruing to i in equilibrium,
ϕ∗i (G)∑n
j=1 ϕ∗j (G)
=Γβ,γi (G)
∑nj=1 Γ
β,γj (G)
, (5)
where
Γβ,γi (G) ≡
1
1 + βd+i − γd−i, and Λβ,γ (G) ≡ 1−
1∑n
i=1 Γβ,γi (G)
. (6)
Γβ,γi (G) is a measure of the local hostility level capturing the externalities associated with agent
i’s first-degree alliance and enmity links. Both Γβ,γi (G) and Λβ,γ(G) are decreasing with β, and
increasing with γ (see Lemma 3 in Online Appendix B). Note also that equation (1) implies thatthe share of the prize accruing to agent i increases in the number of her allies and decreases in thenumber of her enemies.
Next, we characterize the equilibrium fighting effort, and show how it depends on the structureof the network.
5Nor do we impose any non-negativity constraint on xi, given the linearity of the pay-off function. The zero effortlevel is a matter of normalization. One could rewrite the model by replacing the effort level xi with (x + xi). Allresults would be unchanged.
6
Proposition 1. Assume that β + γ < 1/max{λmax(G+), d−max}, where λmax (A
±) denotes thelargest eigenvalue associated with the matrix A±. Assume, in addition, that condition (3) holds
true. Let Γβ,γi (G) and Λβ,γ (G) be defined as in equation (6), and let
cβ,γ (G) ≡(In + βA+ − γA−
)−1Γβ,γ (G) (7)
be a centrality vector, whose generic element cβ,γi (G) describes the centrality of agent i in thenetwork G. Then, under an appropriate restriction of the strategy space, S = [x1,∞) × [x2,∞) ×...× [xn,∞) ⊂ R
n, where xi > −∞ ∀i = 1, 2, . . . , n, there exists a unique interior Nash equilibriumof the n–player simultaneous move game with payoffs given by equation (1), agents’ OPs in equation(2), where the equilibrium effort levels are characterized by
x∗i (G) = V Λβ,γ (G)(1− Λβ,γ (G)
)cβ,γi (G) (8)
for all i = 1, . . . , n, where x∗i (G) > xi. Moreover, the equilibrium OP levels are given by equation(4), and the equilibrium payoffs are given by
π∗i (G) = πi (x∗, G) = V (1− Λβ,γ(G))
(Γβ,γi (G)− Λβ,γ (G) cβ,γi (G)
). (9)
The inequality β + γ < 1/max{λmax(G+), d−max} is a sufficient condition for the matrix in (7)
to be invertible. Equation (9) follows from the set of FOCs. Condition (3) ensures that, for alli = 1, . . . , n, agent i’s payoff, conditional on x∗
−i, is locally concave, implying that the FOCs pindown a local maximum for all players.6 Formally, this guarantees that the strategy profile satisfying(9) is a local Nash equilibrium (Alos-Ferrer and Ania 2001).7 To ensure that this strategy profile is,in addition, a Nash equilibrium in the standard sense, we must impose a lower bound on the statespace to ensure in turn that, conditional on x∗
−i, x∗i (G) is a best reply over the entire admissible
region, [xi,∞).8
The centrality measure, cβ,γi (G) , plays a key role in Proposition 1. Note, in particular, thatthe relative fighting efforts of any two agents equals the ratio between the respective centrality inthe network:
x∗i (G)
x∗j (G)=cβ,γi (G)
cβ,γj (G).
2.3 The Case of Small Externalities
While cβ,γi (G) depends, in general, on the entire network structure, it is instructive to considernetworks in which the spillover parameters β and γ are small. In this case, our centrality measurecan be approximated by the the sum of (i) the Bonacich centrality related to the network of enmities,G−, (ii) the (negative-parameter) Bonacich centrality related to the network of alliances, G+, and(iii) the local hostility vector, Γβ,γ(G).9
6In the rest of this section, we assume that the two conditions under which the Proposition is proven are satisfied.In the empirical analysis, we will check that the estimates of β and γ and the adjacency matrix ensure that the FOCscharacterize an interior Nash equilibrium.
7A local Nash equilibrium is a strategy profile such that no player has an incentive to operate a “small” deviation.A formal definition is provided in Definition 2 in Appendix A. The lower bound on the state space ensures that, atthe equilibrium, no player has an incentive to operate any feasible deviation.
8In Remark 1 in Appendix A we show that it is possible to compute a common lower bound in Proposition 1,x, on the effort levels xi, such that πi(x
∗−i, xi, G) − πi(x
∗, G) < 0 holds for all i = 1, . . . , n if x ∈ [x,∞)n and thatx < x∗
i for all i = 1, . . . , n.9See the Online Appendix D for a more detailed discussion of the Bonacich centrality. A discussion of the Bonacich
centrality with a negative parameter can be found in Bonacich (2007) .
7
Lemma 1. Assume that the conditions of Proposition 1 are satisfied. Then, as β → 0 and γ → 0,the centrality measure defined in equation (7) can be written as
cβ,γ(G) = b·(γ,G−) + b·(−β,G
+)− Γβ,γ(G) +O (βγ) ,
where O (βγ) involves second and higher order terms, and the (µ-weighted) Bonacich centrality withparameter α is defined as b·(α,G) ≡ bΓβ,γ(G)(α,G) = (In − αA)−1
Γβ,γ(G) =∑∞
k=0 αkAkΓβ,γ(G).
Lemma 1 states that the centrality cβ,γ(G) can be expressed as a linear combination of theweighted Bonacich centralities b·(γ,G
−), b·(−β,G+) and the vector Γβ,γ(G). Each Bonacich
centrality gauges the network multiplier effect attached to the system of enmities and alliances,respectively. In particular, b·(γ,G
−) captures how a group i is influenced by all its (direct andindirect) enemies.10 In the case of of weak network externalities (i.e. when β → 0 and γ → 0), theBonacich centrality can be itself approximated as follows:
b·,i(γ,G−
)= Γβ,γ
i (G) + γn∑
j=1
a−ijΓβ,γj (G) + γ2
n∑
j=1
a−ij
n∑
k=1
a−jkΓβ,γk (G) +O
(γ3),
b·,i(−β,G+) = Γβ,γ
i (G) + (−β)n∑
j=1
a+ijΓβ,γj (G) + (−β)2
n∑
j=1
a+ij
n∑
k=1
a+jkΓβ,γk (G) +O
(β3).
Thus, Lemma 1 suggests that, when higher order terms can be neglected, our centrality measure isincreasing in γ and in the number of first order and second order enmities, whereas it is decreasingin β and in the number of first order alliances. Second order alliances have instead a positive effecton the centrality measure.11
Moreover, we can also obtain a simple approximate expression for the equilibrium efforts andthe payoffs in Proposition 1.12
Lemma 2. As β → 0 and γ → 0, the equilibrium effort and payoff of agent i in network G can bewritten as
x∗i (G) = V(Aβ,γ(G) −B
(βd+i − γd−i
))+O (βγ) ,
π∗i (G) = V(Cβ,γ(G) +D
(βd+i − γd−i
))+O (βγ) ,
where Aβ,γ(G), B,Cβ,γ(G) and D are positive constants with Aβ,γ(G) and Cβ,γ(G) being of theorder of O(β) +O(γ).
10The Bonacich centrality measure related to the network of hostilities, b·,i(
γ, G−)
, measures as the local hostilitylevels along all walks reaching i using only hostility connections, where walks of length k are weighted by thegeometrically decaying hostility externality γk (see also the Online Appendix D). Due to the approximation, we onlyconsider links up to order two.
11The intuition for this property is as follows. If i is allied with j and j is allied with k, an increase in the fightingeffort of k reduces the fighting effort of j and this, in turn, increases the fighting effort of i. Consider, instead, thecase in which i is an enemy of j and j is an enemy of k. Then, an increase in the fighting effort of k increases thefighting effort of j and this, in turn, increases the fighting effort of i.
12See the proof of Lemma 2 in Appendix A for the explicit expressions for the constants Aβ,γ(G), B,Cβ,γ(G) andD.
It is useful to note that, when β = γ = 0, then Λβ,γ(G) = 1− 1n, and cβ,γi (G) = 1. Then, the equilibrium expressions
in Proposition 1 simplify to x∗i (G) = V (n− 1)/n2 and π∗
i (G) = V/n2 which are the standard solutions in the TullockCSF.
8
Lemma 2 shows that, when network externalities are small, an agent’s fighting effort increasesin the weighted difference between the number of enmities (weighted by γ) and alliances (weightedby β), i.e. the net local externalities d+i β − d−i γ. The opposite is true for the equilibrium payoff,which is increasing in d+i β − d−i γ. Thus, ceteris paribus, an increase in the spillover from alliances(enmities), parameterized by β (γ), and an increase in the number of allies (enemies) decreases(increases) agent i’s fighting effort and increases (reduces) its payoff. Intuitively, an agent withmany enemies tends to fight harder and to appropriate a smaller share of the prize, whereas anagent with many friends tends to fight less and to appropriate a large size of the prize.
One must remember, however, that this simple result may fail to hold true if β and γ are notsmall and higher-degree links have sizeable effects. In the empirical analysis below, we find thathigher order terms cannot be ignored to draw both positive and normative implications.
2.4 Rent Dissipation and Key Player
In this section, we discuss normative implications of the theory. Our welfare measure is (minus)the total rent dissipation. Since V is exogenous, the rent dissipation equals the total equilibriumfighting effort:
RDβ,γ(G) ≡1
V
n∑
i=1
x∗i (G) = Λβ,γ(G)(1 − Λβ,γ(G))
n∑
i=1
cβ,γi (G).
Since∑n
i=1 π∗i (G) = V
(1− RDβ,γ(G)
), minimizing rent dissipation is equivalent to maximizing
welfare.Next, we define the key player to be the agent whose removal triggers the largest reduction in
rent dissipation (cf. Ballester, Calvo-Armengol and Zenou 2006).
Definition 1. Let G\{i} be the network obtained from G by removing agent i, and assume thatthe conditions of Proposition 1 hold. Then the key player i∗ ∈ N ≡ {1, . . . , n} ∪ ∅ is defined by
i∗ = argmaxi∈N
{RDβ,γ (G)− RDβ,γ (G\{i})
}. (10)
Note that the welfare difference RDβ,γ (G)−RDβ,γ (G\{i}) can be interpreted as the maximumcost that a benevolent policy maker is willing to pay to induce (or force) agent i not to participatein the contest. In Proposition 2 in the Online Appendix B, we show that the identity of the keyplayer is related to the centrality measure defined in equation (7).13
2.5 Examples
In this section, we discuss two types of examples that illustrate general properties of the model.We consider, first, a regular graph where all agents have a symmetric position in the network. Thisgraph is useful for showing how the number of alliances and enmities affects rent dissipation. Then,we consider a path graph with a core-periphery structure that highlights the importance of thecentrality of agents in the network.
13The key player identified in equation (43) in Proposition 2 in the Online Appendix B differs significantly fromthe one introduced in Ballester, Calvo-Armengol and Zenou (2006). Our key player formula is more involved due tothe non-linearity inherent in the contest success function in the agents’ payoffs.
9
Figure 1: The figure shows three examples of regular graphs Gk+,k− : The left panel shows a regulargraph with k+ = k− = 1 (cycle), the middle panel shows a regular graph with k+ = k− = 2 (alattice with periodic boundary conditions, i.e. a torus), and the right panel a Cayley graph withk+ = 1 and k− = 2. Alliances are indicated with thick lines while enmities are indicated with thinlines.
2.5.1 Regular Graph
A regular network, Gk+,k−, has the property that every agent i has d+i = k+ alliances and d−i = k−
enmities. Regular graphs are tractable and enable us to perform comparative statics on the numberof alliances or enmities in the network. Three examples of regular graphs are displayed in Figure1.
Given the symmetric structure, there exists a symmetric Nash equilibrium such that all agentsexercise the same effort. Moreover, ϕ∗
i = ϕ∗ = 1/n for all i = 1, . . . , n, implying an equal division ofthe pie. Under the conditions of Proposition 1, the equilibrium effort and payoff vectors are given,respectively, by:
x∗(k+, k−
)≡ x∗i
(Gk+,k−
)=
(1
1 + βk+ − γk−−
1
n
)×V
n, (11)
π∗(k+, k−
)≡ π∗i
(Gk+,k−
)=
1 + (1 + n)(βk+ − γk−)
n(1 + βk+ − γk−)×V
n. (12)
Standard differentiation implies that x∗ is decreasing in k+ and increasing in k−, whereas π∗ isincreasing in k+ and decreasing in k−. Intuitively, alliances (enmities) reduce (increase) effort andrent dissipation by decreasing (increasing) the marginal return of individual fighting effort. Thisbasic intuition gets confounded in general networks due to the asymmetries in higher-order links.However, the result is unambiguous in regular graphs.
The regular graph nests three interesting particular cases. First, if β = γ = 0, we have astandard Tullock game, with RD0,0
(Gk+,k−
)= (n− 1) /n. Second, consider a complete network
of alliances (k+ = n − 1), where, in addition, β → 1. Then, x∗ → 0 and RD1,γ (Gn−1,0) → 0,i.e., there is no rent dissipation. Namely, the society attains peacefully the equal split of thesurplus, as in Rousseau’s harmonious society. The lack of conflict does not stem here from socialpreferences or cooperation, but from the equilibrium outcome of a non-cooperative game betweenselfish individuals. The crux is the strong fighting externality across allied agents, which takes themarginal product of individual fighting effort down to zero. Third, consider, conversely, a society inwhich all relationships are hostile, i.e., k− = n− 1. Then, RDβ,γ (G0,n−1) → 1 as γ → 1/ (n− 1)2:all rents are dissipated through fierce fighting and total destruction, as in Hobbes’ homo hominilupus pre-contractual society.
10
5 4 1 2 3
Figure 2: The figure shows a path graph, P5, with five agents.
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070.16
0.17
0.18
0.19
0.20
x∗
γ
x∗1
x∗2
x∗3
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0.010
0.015
0.020
0.025
0.030
0.035
0.040
π∗
γ
π∗1
π∗2
π∗3
0.0 0.1 0.2 0.3 0.4
0.00
0.05
0.10
0.15
x∗
β
x∗1
x∗2
x∗3
0.0 0.1 0.2 0.3 0.40.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18π∗
β
π∗1
π∗2
π∗3
Figure 3: The figure shows the equilibrium efforts (left panels) and payoffs (right panels) as functionsof γ and β for two path graphs in which there are only enmity links (upper panels) and only alliancelinks (lower panels), respectively, for V = 1.
2.5.2 Path Graph
Next, we consider a path (core-periphery) graph, P5, with five agents, see Figure 2. No link meansa neutral relationship. Suppose, first, that all links in Figure 2 are enmities. The upper left andupper right panel of Figure 3 show, respectively, effort levels and payoff for different values of γ.The ranking of the effort level yields x∗1(P5) > x∗2(P5) = x∗4(P5) > x∗3(P5) = x∗5(P5).
14 Intuitively,effort is proportional to the centrality in the network. The ranking of pay-offs is opposite: welfareincreases as one moves from the center to the periphery. Fighting effort and payoffs are increasingand decreasing in γ, respectively.
Consider, next, the polar opposite case in which all links in Figure 2 are alliances. The bottomleft and right panels of Figure 3 show, respectively, effort levels and payoffs for different values of β.
14More formally, we have: x∗1(P5) = V
2(γ(γ+2)−2)(γ(3γ2+γ−1)−1)(5−7γ)2(3γ2−1)
, x2(P5)∗ = x∗
4(P5) = V2((γ(3γ+5)−9)γ2+2)
(5−7γ)2(1−3γ2)and
x∗3(P5) = x∗
5(P5) = V 2(γ(γ+2)−2)((γ−1)γ(3γ+1)+1)
(5−7γ)2(3γ2−1).
11
5 4 1 2 3
Figure 4: The figure shows three different graphs, P4, P2 ∪ P2 and P3 ∪ P1, obtained from a pathgraph, P5, due to the removal of an agent.
The ranking of effort yields now, x∗2(P5) = x∗4(P5) < x∗1(P5) < x∗3(P5) = x∗5(P5).15 For a low range
of β’s, the peripheral agents, 3 and 5, exert the highest effort and earn the lowest payoff. However,for higher β’s, agents 3 and 5 earn a higher payoff than agent 1. Note that agents 2 and 4 exertthe lowest effort, in spite of agent 1 being the most central player. The reason is that agents 2 and4 are connected, respectively, to agents 3 and 5 who lie at the periphery of the network, and whohave no other connections. Thus, agents 2 and 4 can free ride on the high effort exerted by 3 and5.16 Finally, the efforts and payoffs of agents 1, 3 and 5 are non-monotonic, due to the fact that,when β is high, there is more free riding. Agents 2 and 4 exert very low fighting effort, and inducesubstitution effects (i.e., higher effort) from their respective neighbors.
The results are consistent with Lemma 2. In the upper panel, as γ → 0 the effort is lowest andpayoff is highest for the agent who has the smallest number of enemies (i.e., agent 1). In the lowerpanel, as β → 0 the effort is highest for the agents who have only one ally (i.e., agents 3 and 5).
Finally, we perform a key player analysis. Figure 4 shows the three different graphs, P4, P2∪P2
and P3∪P1, obtained from removing one agent from P5. When all links are enmities, the maximumreduction in rent dissipation is attained by removing the central agent, 1. The reason is that, inthis case, a graph with more walks (hence, stronger network feedback effects) amplifies fighting.Removing the central agent in P5 (as displayed in P2∪P2) yields the largest reduction in the numberof walks, resulting in the lowest total fighting. In contrast, when all links are alliances, fightingis decreasing in the number of walks. In this case, the maximum reduction in rent dissipation isattained by removing a peripheral agent (either 3 or 5).
2.6 Heterogeneous Fighting Technologies
So far, we have maintained that all agents have access to the same technology turning fighting effortinto OP. This was useful for keeping the focus sharply on the network structure. In reality, armedgroups typically differ in size, wealth, access to weapons, etc. In this section, we generalize ourmodel by allowing fighting technologies to differ across players. We restrict attention to additiveheterogeneity, since this is crucial for achieving identification in the econometric model presented
15The analytical expressions are x∗1(P5) = V
2(β2(β+1)(β(3β−10)+5)−2)(7β+5)2(3β2−1)
, x∗2(P5) = V
2(β2(β(5−3β)+9)−2)(7β+5)2(3β2−1)
and x∗3(P5) =
V 2((β−2)β−2)(β(β+1)(3β−1)−1)
(7β+5)2(1−3β2).
16This is related to the interpretation of the Bonacich centrality with a negative parameter. In this case a nodeis more powerful to the degree that its connections themselves have few alternative connections (see Sec. 1.1.1 inBonacich, 2007).
12
below.17
Suppose that agent i’s OP is given by:
ϕi(G,x) = ϕi + xi + βn∑
j=1
a+ijxj − γn∑
j=1
a−ijxj, (13)
where ϕi is a group-specific shifter affecting OP (e.g., its military capability).18
In the appendix, we show that the equilibrium OP is unchanged, and continues to be givenby equation (4). Likewise, equation (5) continues to characterize the share of the prize appro-priated by each agent. Somewhat surprising, the share of resources appropriated by group i,ϕ∗i (G)/
∑nj=1ϕ
∗j(G), is independent of ϕi. However, ϕi affects the equilibrium effort exerted by
group i and its payoff. In particular, the vector of the equilibrium fighting efforts is now given by
x∗ = (In + βA+ − γA−)−1(V Λβ,γ(G)(1 − Λβ,γ(G))Γβ,γ(G) − ϕ), (14)
where the definitions of Λβ,γ(G) and Γβ,γ(G) are unchanged (see Proposition 3 in the OnlineAppendix B).
Equation (13) will be the basis of our econometric analysis where we introduce both observableand unobservable sources of heterogeneity. In particular, time-varying shocks to ϕ will be key forthe econometric identification.
3 Empirical Application - The Second Congo War
In this section, we apply the theory to the recent civil conflict in the Democratic Republic ofCongo (henceforth, DRC). Our goal is to estimate the externality parameters β and γ from thestructural equation (13) characterizing the Nash equilibrium of the model. Equipped with pointestimates of the structural parameters we test key restrictions imposed by the theory and performcounterfactual pacification policies. We start by presenting the historical context of the DRCconflict. Then, we discuss the data sources and how the network structure is observed from thedata. Finally, we proceed to the econometric model and to the discussion of our identification andestimation procedure.
3.1 Historical Context
We study the Second Congo War, sometimes referred to as the ”Great African War”. Detailedaccounts of this conflict can be found in Prunier (2011) and Stearns (2011). The DRC is the largestSub-Saharan African country in terms of area, and is populated by about 75 million inhabitants.It is a classic example of a failed state. After gaining independence from Belgium in 1960, it expe-rienced recurrent political and military turbulence that turned it into one of the poorest countriesin the world, despite its abundance of natural resources (including diamonds, copper, gold andcobalt). The DRC is also a very ethnically fragmented country with over 200 ethnic groups. The
17It is possible to solve for richer forms of heterogenity, such as a multiplicative one. However, we do not focus onsuch alternative cases, since it becomes impossible to identify econometrically the parameters of the model.
18An alternative interpretation is to consider our model as the linear approximation of a logit-form CSF (cf.
Skaperdas, 1996). Consider the payoff function πi : Gn × Rn → R with πi(G,x) ≡ V eλϕi(G,x)
∑
nj=1 e
λϕj(G,x) − xi, for all
i = 1, . . . , n, where ϕi(G,x) is the OP of agent i given by equation (2). In the limit λ → 0 we obtain the ratio form
πi(G,x) = V1+λϕi(G,x)+O(λ2)
n+λ∑
nj=1 ϕj(G,x)+O(λ2)
− xi = Vλ−1+ϕi(G,x)+O(λ)
nλ−1+∑
nj=1 ϕj(G,x)+O(λ)
− xi. Hence, we can introduce the shifted OP
from equation (13) with ϕi = λ−1 for all i = 1, . . . , n.
13
Congo conflict is emblematic of the role of natural resource rents and of the involvement of manyinter-connected domestic and foreign actors. In particular, the conflict involved three Congoleserebel movements, 14 foreign armed groups, and several militias (Autesserre 2008). In such complexand fragmented warfare, alliances and enmities play a major role.
The Congo Wars are intertwined with the ethnic conflicts in neighboring Rwanda and Uganda.The culminating event is the Rwandan genocide of 1994, where the Hutu-dominated governmentof Rwanda supported by ethnic militias such as the Interahamwe, persecuted and killed nearly amillion of Tutsis and moderate Hutus in less than one hundred days. After losing power to the Tutsirebels of the Rwandan Patriotic Front (RPF), over a million Hutus fled Rwanda and found refugein the DRC, governed at that time by the dictator Mobutu Sese Seko. The refugee camps hosted,along with civilians, former militiamen responsible of the Rwandan genocide. These continued toclash with the Tutsi population living both in Rwanda and in the DRC, most notably in the Kivuregion (Seybolt 2000).
As ethnic tensions escalated, a broad coalition centered around Uganda and Rwanda, but alsocomprising several other African states, supported the anti-Mobutu rebellion led by Laurent-DesireKabila’s Alliance of Democratic Forces for the Liberation of Congo (ADFL). The First CongoWar (1996-97) ended with Kabila’s victory. However, the relationship of the new president of theDRC with his former Tutsi allies and his main foreign sponsors, Rwanda and Uganda, deterioratedrapidly. The Second Congo War erupted in 1998 where Kabila received the support of some formerforeign allies (Angola, Chad, Namibia, Sudan and Zimbabwe) and of the Hutu militias that hadpreviously supported Mobutu.19 His main enemies were Uganda, Rwanda and a network of rebelgroups. Many rebel groups were linked to foreign powers: Uganda sponsored the Rally for CongoleseDemocracy - Kisangani (RCD-K), while the Congolese Liberation Movement (MLC), the Rally forCongolese Democracy - Goma (RCD-G) received support from Rwanda (Seybolt 2000). The heartof the war was the Eastern border and the conflict spread fast to the whole country, with manyactors taking part in the conflict out of hostility to other specific actors. These include, amongothers, the anti-Ugandan rebel forces of the Allied Democratic Forces and the Lord’s ResistanceArmy, or the anti-Angolan UNITA forces.
The Second Congo War ended officially in 2003, although fighting is still going on today. Itis the deadliest conflict since World War II, with between 3 and 5 million lives lost (Olsson andFors, 2004; Autesserre, 2008). In spite of the numerous shifts that occurred in 1997, the web ofalliances and enmities remained remarkably stable thereafter. As documented by Prunier (2011:187ff), most alliances and enmities were determined by international and domestic factors of thecountries involved, and remained constant over the entire conflict (at least until 2010, the extentof time covered by our data). Many groups chose their loyalties following Realpolitik and trying tobalance rising neighbors like South Africa or Sudan.
While many groups involved in the Second Congo War took stands vis-a-vis the DRC gov-ernment, there were flagrant violations of transitivity, and the conflict cannot be described as acoherent two-camp (pro- versus anti-DRC government) war. In Prunier’s words, ”the continentwas fractured, not only for or against Kabila, but within each of the two camps” (2011: 187).Recurrent clashes erupted among the main factions - like in the case of the infamous violent clashesbetween RCD-G and RCD-K (Prunier 2011: 240ff; Turner 2007: 200). Even the historical alliancebetween Uganda and Rwanda, which was forged on ethnic and geopolitical grounds cracked a fewtimes, resulting in important clashes between the two armies or between groups sponsored by ei-ther of them (Turner, 2007: 200). Similarly, there was in-fighting among different pro-governmentparamilitary groups, such as the Mayi-Mayi militias, (cf. Prunier 2011: 281). The DRC army
19In 2001, Laurent-Desire Kabila was assassinated, and was later replaced by his son Joseph Kabila.
14
itself was notoriously prone to internal clashes and mutinies, spurred by the fact that its units aresegregated along ethnic lines and often correspond to former ethnic militias or paramilitaries thatgot integrated into the national army (cf. Prunier 2011: 305ff).20 In summary, far from being awar between two unitary alliances, the conflict engaged a complex web of alliances and enmities,with many non-transitive links.
Two other aspects of the conflict are noteworthy. First, with the exception of the DRC armedforces, most actors were active in limited parts of the country. Ethnic militias (i.e., the vastmajority of local armed groups) typically fought in territories adjacent to those where their affiliatedpopulation lived. Second, weather conditions were important in determining the intensity of theconflict in different regions. Figure 5 displays the fighting intensity and average climate conditionsfor different ethnic homelands in the DRC.21 Weather conditions vary considerably both acrossregions of the DRC and over time.
3.2 Data
We build a panel dataset for the period 1998-2010 that includes both the official years of theSecond Congo War (until 2003) and its turbulent aftermath. The unit of observation is at thefighting group×year level (annual frequency). The variables used in the estimations are built froma variety of data sources. We describe these and the construction of the variables in this section.The related summary statistics are displayed in Table 1.
Fighting – Our main data source is the Armed Conflict Location and Events Dataset (ACLED2012).22 This dataset contains 4765 geo-localised violent events taking place in the DRC involving85 fighting groups. For each such event, ACLED provides information on the exact location, thedate and the identities of the involved groups – including information about which groups fight onthe same side and which fight on opposite sides.23 In the recent literature ACLED has been used sofar with the purpose of building geolocalized measurement of violence. Here, besides geolocalization,we also exploit bilateral information about which groups fight together or against each other todocument the fine-grained structure of the network of alliances and enmities. To the best of ourknowledge, our study is the first to exploit this information.
Our main dependent variable is group i’s yearly Fighting Effort. We measure xit as the sumover all ACLED fighting events involving group i in year t. In the robustness section, we construct
20Turner (2007) describes a typical clash between army fractions in 2004: ”In the aftermath of the peace agreementsof Sun City and Pretoria, Congo was supposed to create a unified national army and civil-territorial administration.Some Rwandophone officers of North and South Kivu led the resistance to brassage (intermingling) of officers andtroops from various composants. The two most prominent of these were Colonel Jules Mutebutsi (a Munyamulengefrom South Kivu) and General Laurent Nkunda (Rwandophone Tutsi allegedly from Rutshuru in Kivu). Theseofficers led a mutiny against their superiors, and briefly took over the city of Bukavu (capital of South Kivu) (2007:96).”
21The data used for generating this figure are discussed in detail below.22This a well established data source in the literature. Recent papers using ACLED include, among others, Berman
et al. (2014), Cassar, Grosjean, and Whitt (2013), Michalopoulos and Papaioannou (2013), and Rohner, Thoenig,and Zilibotti (2013b).
23Here are three examples: On the 18th of May 1999 a battle took place between ”RCD: Rally for CongoleseDemocracy (Goma)” and ”Military Forces of Rwanda”, on one side, and the ”Military Forces of DRC” on the otherside. On the 13th of January 2000 there was a battle between ”Lendu Ethnic Militia” and ”Military Forces of DRC”,on the one side, and ”Hema Ethnic Militia” and ”RCD: Rally for Congolese Democracy”, on the other side. Onthe 3rd of February 2000, the ”MLC: Congolese Liberation Movement” together with ”Military Forces of Uganda”confronted the allied forces of the ”Military Forces of DRC” and ”Interahamwe Hutu Ethnic Militia”.
15
Figure 5: Average rainfall and number of violent events (ACLED) in the DRC.
16
Table 1: Summary statistics.
Variable Obs. Mean Std. Dev. Min. Max.
Total Fighting 1190 5.23 22.72 0 300Total Fighting of Enemies 1190 50.48 92.89 0 645Total Fighting of Allies 1190 38.80 74.77 0 493d− (Number Enemies) 1190 2.61 3.59 0 20d+ (Number Allies) 1190 2.24 3.41 0 18Foreign 1190 0.31 0.46 0 1Government Organization 1190 0.20 0.40 0 1Rain fall (t− 1) 1190 125.35 26.36 58.02 197.35
alternative measures of fighting effort by restricting the count to the more conspicuous events suchas those classified by ACLED as battles or those involving fatalities.
Rainfall – For the purpose of our instrumentation strategy we build the yearly average ofrainfall in the homeland of each fighting group. We use gauge-based rainfall measure from theGlobal Precipitation Climatology Centre (GPCC) (Schneider et al. 2011), at a spatial resolution of0.5◦×0.5◦ grid-cells. This dataset is widely used and is renowned for its precision and fine resolution.The homeland of a fighting group corresponds to the spatial zone of its fighting operations (i.e.convex hull containing all geolocalized ACLED events involving that group at any time during theperiod 1998-2010). Then, for each year t, we compute the average rainfall in the grid-cell of thehomeland centroid. Alternative constructions are considered in our robustness analysis.24
One potential concern is that rain-gauges located at the ground level may be damaged by localfighting activities. Therefore, in the robustness section, we also use satellite-based rainfall measures.These are generally less precise, but it is safe to assume that the measurement error is uncorrelatedwith ground-level fighting. We use two alternative annual dataset. The first comes from the GlobalPrecipitation Climatology Project (GPCP) from NOAA and has a spatial resolution of 2.5◦×2.5◦.The second is the Tropical Rainfall Measuring Mission (TRMM) from NASA at a resolution of0.5◦×0.5◦. These dataset use atmospheric parameters (e.g. cloud coverage, light intensity) asindirect and noisy measures of rainfall. Hence both of these satellite-based measures are much lessprecise than the rain-gauge measure.
Covariates – The following variables are used to generate the set of control variables. Gov-ernment Organization is a binary variable equal to 1 for fighting groups that are officially affiliatedto a domestic or foreign government. This amounts to 17 groups out of 85, corresponding to themilitary and police forces of Angola, Burundi, Chad, DRC, Namibia, Rwanda, South Africa, Sudan,Uganda, Zambia and Zimbabwe. The dummy Foreign is equal to 1 for all foreign actors, and 0 forall fighting groups that originate from the DRC. In total 26 groups are coded as Foreign. Finallyfor all groups we compute the yearly amount of fighting events in which they are involved outsidethe DRC. The resulting variable, Fighting Effort Outside DRC, is used as proxy for the global scopeof operation of a group.
24One option consists in averaging rainfall across all grid-cells of the homeland (not only at the centroid). As foralternative construction of the homeland we consider an ellipsoid based on a subset of ACLED events (i.e. withinone standard-deviation) or an homeland based on the ethnic affiliations of the fighting group. The main results arereported in the robustness section.
17
3.3 The Fighting Network
We estimate the network of alliances and enmities using two data sources. First, we use theYearbook of the Stockholm International Peace Research Institute (SIPRI, see Seybolt, 2000).Second, we use the dyadic information provided by ACLED. Details are provided below.
As our primary criterion, we follow the classification provided by SIPRI, which lists alliancesbetween the major actors, including both actors operating in the same region and groups fightingin different parts of the country but supporting each other logistically. The limitation of theSIPRI data is that they do not cover small armed groups and militias nor do they contain detailedinformation about bilateral enmity links.
For this reason, we also use the dyadic information provided by ACLED. In particular, we codetwo groups (i, j) as allies (i.e. a+ij = a+ji = 1) if they have been observed fighting on the sameside in at least one occasion during the sample period, and if, in addition, they have never beenobserved fighting on opposite sides. Conversely, we code two groups as enemies (i.e. a−ij = a−ji = 1)if they have been observed fighting on opposite sides on at least two occasions, and they have neverbeen observed fighting on the same side.25 We code all other dyads as neutral (i.e. a+ij = a−ij = 0).
Slightly less than 3% of the dyads are coded as allies, and slightly more than 3% are coded asenemies. The remaining dyads are classified as neutral, either because they were never involvedjointly in any fighting event (93% of the dyads), or because of ambiguities, namely, the two groupswere recorded fighting sometimes on the same side, and sometimes on opposite sides. The propor-tion of all dyads that are classified as neutral due to such an ambiguity is less than 1%. On averagea group has 2.6 enemies and 2.2 allies (see Table 1).
Our methodology may raise three concerns. First, we assume a time-invariant network. Onemight fear that alliances and enmities get reshuffled throughout the war. From a historical per-spective it appears as if many changes in the system of alliances took place at the end of theFirst Congo War, but that alliances remain broadly stable thereafter (cf. Prunier 2011). However,one might worry that this broad pattern may not apply to small groups, whose behavior is notequally well-documented by the conflict literature. To get a sense of the potential importance ofthe problem, we search for instances in which there is a clear switch in the nature of a dyadic re-lationship in ACLED. A clear evidence of structural change would be that there exists a thresholdyear T ∈ (1998 − 2010) such that two groups would be classified as allies (enemies) if one restrictedattention to the years up to T, while the same two groups would be classified as enemies (allies) ifone considered the years after T. We have eight dyads, a mere 0.2% of the total 3570 dyads thatconform with this pattern. This is a strong indication that in most of the 1% dyads for whichan ambiguity arises, this ambiguity is due to occasional clashes between troops (or one-shot tac-tical alliances within a single battle) rather than to structural changes in system of alliances andenmities. This is reassuring for our assumption that the network is time-invariant.26
The second concern is that the construction of the network partly relies on the same ACLEDdata that we use to measure the outcome variable (i.e fighting). Here, let us emphasize twoimportant differences. First, for the network we exploit the bilateral (dyadic) information whichis not used to construct the outcome variable. Second, the network is time-invariant, whereasour econometric analysis exploits the time variations in fighting efforts, controlling for group fixedeffects, as discussed in more detail below. To further alleviate this concern, in Section 4 we build
25Given that in our theoretical setting all groups are competing, by definition, for the same prize, we require atleast two instances of fighting against each other to code two groups as enemies. As shown below, our results arerobust to alternative coding where groups are enemies when they fight against each other in at least one instance.
26To further alleviate this concern, in the robustness section we restrict the sample to 1998-2007, since there isanedoctal evidence that after 2007 some militias were wiped out or absorbed by the DRC Army. The results arerobust.
18
LRA
UNITA
FAA/M...
Uganda
Hutu
Burundi J. Kab.
SPLA/M
Sudan
FNL
L. Kab.
ADF
RCD G...
FDLR
Rwand...
Mayi-...
CNDD-...
Zimba...
Chad
Hutu
RCD K.
CNDP
MLC
DRC M...
UPC
Lendu ADFL
RCD
Namibia
FNI
Rwand...
FPJC
FRPI
Hema
Mutiny
ALIR
MRC
NALU
FLC
South...
DRC P...
PARECO
PUSIC
Rwand...
UPPS
BDK
Zambia
RUD
Enyele
Lobala
FRF
APCLS
Lendu
Un. M... F. DRC
Haut-...
Mayi-...
Mayi-...
Minem...
Bomboma
Masunzu
Wager...
CNDD-...
RCD N...
Figure 6: The figure shows the network of alliances and conflicts in the DRC. Thick lines indicatealliances, while thin lines indicate enmities. The nodes’ colors and sizes represent their centrality(cf. Section 2.2).
the network using SIPRI and ACLED data from the period 1998-2001, while the outcome variableuses the information for the period 2002-2010. This comes at a high cost in term of informationloss since the conflict attains its highest intensity in the initial years. The results are robust, albeitless precisely estimated.
Third, we are likely to miss some network links. It is likely that we code as neutral some dyadsthat are in fact allies or enemies but did not participate into common fighting events (e.g. due tospatial distance). Such missing links create measurement errors that can bias the estimates of thefighting externalities (Chandrasekhar and Lewis, 2011). The strategy to tackle this issue is twofold.On the one hand, we check the robustness of our results to alternative coding rules of alliances andenmities.27 On the other hand, we perform Monte Carlo simulation to assess the bias associatedwith an imperfect observation of missing links.
Figure 6 illustrates the network of alliances and rivalries in DRC. Not surprisingly, the armedforces of DRC have the highest centrality. The other groups with a high centrality are the Rallyfor Congolese Democracy (Goma and Kisangani), the Mayi-Mayi militias, and the foreign armiesof Rwanda, Uganda, Zimbabwe, Namibia, Sudan, and Angola. All these groups are known to haveplayed a salient role in the Second Congo War.
27We consider a variety of alternative coding rules of alliances and enmities. In addition, we check the robustnessof the results to using alternative data sources. In particular, we supplement our information with that provided inthe ”Non-State Actor Data” of Cunningham, Gleditsch and Salehyan (2013).
19
3.4 Econometric Model
The basis of our empirical analysis is Section 2.6, allowing for exogenous sources of heterogeneityin the OP of groups. Equation (13) can be turned into an econometric equation by assuming thatthe individual shocks ϕi comprise both observable and unobservable shifters. More formally, weassume that ϕi = z′iα + ǫi, where zi is a vector of group-specific observable characteristics, andǫi is an unobserved shifter. Replacing xi and ϕi by their respective equilibrium values, yields thefollowing structural equation:
x∗i = ϕ∗i (G)− β
n∑
j=1
a+ijx∗j + γ
n∑
j=1
a−ijx∗j − z′iα− ǫi. (15)
It is important to recall that ϕ∗i is fully characterized by the externality parameters and the time-
invariant network structure, (β, γ, d−i , d+i ) , and is independent of the realizations of individual
shocks (zi, ǫi) (see equation 4 and Section 2.6). Our goal is to estimate the network parameters βand γ. The estimation is subject to a simultaneity or reflection problem (Manski 1993; Boucheret al. 2012), a common challenge in the estimation of network externalities. A related issuein this class of models is that it is difficult to separate contextual effects, i.e., the influence ofplayers’ characteristics, from endogenous effects, i.e., the effect of outcome variables via networkexternalities. In our model, the endogenous effect is associated with the fighting effort exerted bya group’s allies and enemies. Although our theory postulates no contextual effect, it is plausiblethat omitted variables affecting x∗i are spatially correlated, implying that one cannot safely assumespatial independence of ǫi. Ignoring this problem might yield inconsistent estimates of the spilloverparameters.
The reflection problem can be tackled by using instruments to obtain consistent estimatesof the spillover effects. For instance, in a recent study on public good provision in a networkof Colombian municipalities, Acemoglu, Garcia-Jimeno and Robinson (2014) use as instrumentshistorical characteristics of local municipalities which are argued to be spatially uncorrelated. Inour case, it is difficult to single out time-invariant group characteristics that affect the fightingefforts of a group’s allies or enemies without invalidating the exclusion restriction. For instance,cultural or ethnic characteristics of group i are likely to be shared by its allies. For this reason, wetake the alternative route of identifying the model out of exogenous time-varying shifters affectingthe fighting intensity of allies and enemies over time. This panel approach has the advantagethat we can difference out any time-invariant heterogeneity, thereby eliminating the problem ofcorrelated fixed effects.
Panel Specification – We maintain the assumption of an exogenous time-invariant network,and assume that the conflict repeats itself over several years. We abstract from reputational effects,and regard each period as a one-shot game. These are strong assumptions, but they are necessary toretain tractability. The variation over time in conflict intensity is driven by the realization of group-and time-specific shocks, amplified or offset by the endogenous response of the players which, inturn, hinges on the network structure. More formally, we allow both x∗i and ϕi to be time-varying:
ϕit = z′itα+ ei + ǫit. (16)
Here, zit is a vector of observable shocks with coefficients α, ei is a vector of (unobservable) time-invariant group-specific characteristics, and ǫit is a i.i.d., zero-mean unobservable shock. Rainfallmeasures are examples of observable shifters zit that will be key for identification. The panelanalogue of equation (15) can then be written as:
Fightit = fei − β × Fightall
it + γ × Fightene
it − z′itα− ǫit. (17)
20
where
Fightit = x∗it
Fightall
it =
n∑
j=1
a+ijx∗jt,
Fightene
it =
n∑
j=1
a−ijx∗jt,
fei = −ϕ∗i (G)− ei. (18)
The panel dimension allows us to filter out any time-invariant correlated effects by includinggroup fixed effects fei. However, due to the reflection problem discussed above, the two covariatesFight
all
it (Total Fighting of Allies) and Fightene
it (Total Fighting of Enemies) are correlated withthe error terms. In other words, there is an endogeneity problem because the effort of group i’sallies and enemies are affected by group i’s effort. Thus, OLS estimates are inconsistent.
The problem can be addressed by an instrumental variable strategy. This requires identifyingexogenous sources of variations in the fighting efforts of group i’s allies and enemies that do notinfluence group i’s fighting effort directly. To this aim, we use time-varying climatic shocks (rainfall)impacting the homelands of armed groups. In line with the empirical literature and historical casestudies (Dell 2012), we focus on local rainfall as a time-varying shifter of OP, and hence thefighting effort of allies and enemies. We expect groups affected by positive rainfall shocks to fightless, since local rainfall increases the agricultural surplus, thereby pushing up the reservation wagesof productive labor and, hence, the opportunity cost of fighting. This channel linking rainfall toconflict has been documented, among others, by Jia (2014), Hidalgo et al. (2010), Miguel et al.(2004), Vanden Eynde (2011).
To be a valid instrument, rainfall in the homelands of the allies (enemies) must be correlatedwith the allies’ (enemies’) fighting efforts. We show below that this is so in the data. The exclusionrestriction requires that rainfall in the homelands of group i’s allies and enemies have no directeffect on fighting efforts of group i. Although rainfall is likely to be spatially correlated, due to theproximity of the homelands of allied or enemy groups, this is not a problem since we control forthe rainfall in group i’s homeland in the second-stage regression. For instance, suppose that groupi has a single enemy, group k, and that the two groups live in adjacent homelands. Rainfall in k’shomeland is correlated with rainfall in i’s homeland. However, rainfall in k’s homeland is a validinstrument for k’s fighting effort, as long as rainfall in i’s homeland is included as a non-excludedinstrument. A potential issue arises if rainfall is measured with error, and measurement error hasa non-classical nature. We tackle this issue below in the robustness analysis.
Potential violation of the exclusion restriction could arise if there were intense within-countrytrade. For instance, a drought destroying crops in Western Congo could translate into strong priceincreases of agricultural products throughout the entire DRC, thereby affecting fighting in theEastern part of the country. Such a channel may be important in a well-integrated country withlarge domestic trade. Yet, in a very poor country like the DRC with a disintegrating government,very lacunary transport infrastructure and a disastrous security situation, effective transport costsare high and inter-regional trade is limited for most goods. The result is a very localized economydominated by subsistence farming. Hence, spillovers through trade are unlikely to be large.
21
3.5 Estimates of the Fighting Externalities
We now present the estimates of the system of structural equations (17) based on our panel of 85armed groups over 1998-2010. In all specifications, we include group fixed effects and year dummies,and cluster standard errors at the group level.
Table 2 displays the baseline results. Column (1) starts with an OLS specification. We controlfor current and lagged rainfall at the centroid of group’s homeland in a flexible way, allowing forboth linear and quadratic terms. Total Fighting of Enemies increases a group’s fighting effort,whereas Total Fighting of Allies decreases it. These results conform with the prediction of thetheory, although the coefficient of Total Fighting of Allies is not statistically significant.
In column (2) we add a set of time-varying control variables by assuming that common (unob-served) shocks have heterogeneous effects across groups that differ by some observable characteris-tics. More specifically, we build three time-invariant group characteristics and interact each of themwith year dummies. The first characteristic corresponds to a binary variable capturing whether agroup is a government organization (as opposed to a non-state actor). State organizations benefitfrom stable funding and legitimacy conveyed by the law, and thus tend to be larger and betterequipped than local militias and rebel groups. For these reasons, government armies are likely tobe affected differently from informal armed groups by global economic and political shocks. Thetwo other characteristics are proxies for groups’ size.28 Both are binary variables coding, respec-tively, for groups with at least 10 enemies, and for groups involved in at least 20 violent eventsper year outside of the DRC (each variable captures groups between the top 5-10 percent of thesample). The results are not sensitive to using different thresholds or to dropping either of them.The estimates of column (2) are very similar to those of column (1).
Due to the reflection problem discussed above, OLS estimates are inconsistent. Therefore, in thenext columns, we replicate specifications of columns (1)-(2) in a 2SLS setup using first a restrictedset of instruments (columns 3 and 4), and then an enlarged one (columns 5 and 6). The related firststage regressions are reported in Table 3, where, for presentational purposes, only the coefficients ofthe excluded instruments are displayed. Remarkably, there is a stable pattern across all first-stageregressions by which rainfall in the enemies’ homelands affects negatively the fighting effort of theenemy groups (and not that of allied groups), whereas rainfall in the allies’ homelands decreasesthe fighting effort of the allied group (and not that of the enemies). This pattern conforms withthe theoretical predictions.
In Columns (3) and (4) of Table 2, Total Fighting of Enemies and Total Fighting of Allies areinstrumented, respectively, by the one-year lag of average rainfall in the homelands of enemies andallies, including both a linear and a quadratic term in the first-stage regressions. In column (3),the coefficients of the variables of interest have the expected sign and are significant at the 5%level. The coefficients are larger than in the OLS regressions, suggesting that the non-instrumentedspecification suffers from an under-estimation bias due to the omission of the network externalities.Column (4) includes the large set of time-varying controls. This leads to a weak instrument problemand the coefficient of Total Fighting of Enemies turns insignificant.
Columns (1)-(4) of Table 3 display the first stage regressions for Total Fighting of Enemies andTotal Fighting of Allies corresponding to columns (3)-(4) in Table 2. We see that the instrumentshave statistically significant coefficients, with the expected sign, but their overall statistical poweris borderline in absence of time-varying controls (with a Kleibergen-Paap F-stat equal to 9.34)and becomes weak once controls are included (F-stat equal to 6.65). This issue is solved belowby expanding the set of instruments. The null hypothesis of the Hansen J test is not rejected
28We have no good data for troop size. Such data exist only for a small subset of the groups, see the InternationalInstitute of Strategic Studies (IISS) or the Small Arms Survey (SAS).
22
Table 2: Baseline regressions (second stage).
Dependent variable: Total Fighting
(1) (2) (3) (4) (5) (6)
Tot. Fight. Enemies 0.09*** 0.06*** 0.12** 0.08 0.14*** 0.09**(0.02) (0.02) (0.05) (0.07) (0.05) (0.05)
Tot. Fight Allies -0.01 -0.02 -0.16** -0.17** -0.15* -0.14**(0.02) (0.01) (0.07) (0.08) (0.08) (0.07)
Group FE, annual TE, rain controls Yes Yes Yes Yes Yes YesAdditional controls No Yes No Yes No YesEstimator OLS OLS IV IV IV IVSet of Instrument Variables n.a. n.a. Restricted Restricted Full FullObservations 1190 1190 1190 1190 1105 1105R-squared 0.340 0.415 0.241 0.323 0.254 0.364
Notes: An observation is a given armed group in a given year. The panel contains 85 armed groups between1998 and 2010. Robust standard errors allowed to be clustered at the group level in parentheses. Significancelevels are indicated by * p < 0.1, ** p < 0.05, *** p < 0.01.
(with p-value of 0.53) indicating that the overidentification restrictions are valid. Interestingly, thefighting effort of enemies is highly correlated with rainfalls in enemies homeland while the totalfighting effort of the allies is highly correlated with the rainfall in allies homeland. This reassuringpattern is robust across all columns of Table 3. Moreover, consistent with the findings of literaturediscussed above, a higher rainfall in period t − 1 predicts a lower fighting effort. This pattern isconfirmed in specifications including both the current and lagged rainfall.
In column (5) of Table 2, we consider a larger set of instruments comprising current-yearrainfall (linear and quadratic terms) as well as current and lagged rainfalls of degree 2 neighbors(i.e. enemies of enemies and of enemies of allies).29 The second-stage coefficients are stable andsignificant. The statistical power of the first stage (columns (5)-(6) in Table 3) is now high abovethe conventional threshold of 10 (F-stat at 19.16).
Column (6) of Table 2 – our preferred specification – includes the additional time-varyingcontrols and uses the expanded set of instruments. The two coefficients of interest have the expectedsigns and are statistically significant at the 5% level. The statistical power of the instruments inthe first stage (columns (7)-(8) in Table 3) is large, with a Kleibergen-Paap F-statistic equal to13.71. The estimates of the fighting externalities are quantitatively large. A one standard deviation(s.d.) increase in Total Fighting of Enemies (93 violent events) translates into a 0.37 s.d. increasein the Fighting Effort of the group (+8.4 violent events). A one s.d. increase in Total Fighting ofAllies (75 violent events) translates into a 0.46 s.d. decrease in the Fighting Effort of the group(-10.5 violent events). We also check that, conditional on estimates of β and γ, the second-ordercondition, (3), holds true for all groups in conflict. This guarantees that we estimate an interiorequilibrium where the invertibility condition of Proposition 1 is also satisfied.
29When we use the current and past average rainfall in enemies’ and allies’ homelands as instruments, we alsocontrol for the current and past average rainfall in the own group homeland in the second-stage regression. Thisis important, since the rainfall in enemies’ and allies’ homelands is correlated with the rainfall in the own grouphomeland. Omitting the latter would lead to a violation of the exclusion restriction.
23
Table 3: Baseline regressions (first stage).
IV regression of column (3) IV regression of column (4) IV regression of column (5) IV regression of column (6)
Dep. Variable: Tot.Fi. Enemy Tot.Fi.Allied Tot.Fi. Enemy Tot.Fi.Allied Tot.Fi. Enemy Tot.Fi.Allied Tot.Fi. Enemy Tot.Fi.Allied
(1) (2) (3) (4) (5) (6) (7) (8)
Rain (t-1) Enemies -1.45*** 0.03 -1.30*** 0.07 -1.34*** 0.18 -1.20*** 0.15(0.23) (0.10) (0.27) (0.09) (0.24) (0.11) (0.30) (0.12)
Sq. Rain (t-1) Ene. 0.00*** 0.00 0.00*** 0.00 0.00*** 0.00 0.00*** 0.00(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
Rain (t-1) Allies -0.01 -1.02*** -0.03 -0.99*** -0.26 -0.74*** -0.32 -0.73***(0.22) (0.15) (0.18) (0.14) (0.29) (0.19) (0.29) (0.19)
Sq. Rain (t-1) Alli. 0.00 0.00*** 0.00 0.00*** 0.00 0.00*** 0.00 0.00***(0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)
Current Rain Enemies -0.97*** 0.10 -0.99*** 0.08(0.25) (0.13) (0.27) (0.13)
Sq. Curr. Rain Ene. 0.00*** -.00 0.00 0.00(0.00) (0.00) (0.00) (0.00)
Current Rain Allies -0.21 -0.67*** -0.30 -0.66***(0.18) (0.12) (0.20) (0.12)
Sq. Curr. Rain Alli. 0.00 0.00*** 0.00* 0.00***(0.00) (0.00) (0.00) (0.00)
Current rain enemies of enemies -0.11*** -0.05* -0.10** -0.04(0.04) (0.03) (0.04) (0.03)
Sq. current rain enemies of enemies 0.00 0.00 0.00 0.00(0.00) (0.00) (0.00) (0.00)
Current rain enemies of allies -0.12 -0.03 -0.08 -0.04(0.08) (0.04) (0.08) (0.04)
Sq. current rain enemies of allies 0.00* 0.00 0.00 0.00(0.00) (0.00) (0.00) (0.00)
Rain enemies of enemies (t-1) -0.15*** -0.06** -0.19*** -0.07**(0.05) (0.03) (0.05) (0.03)
Sq. rain enemies of enemies (t-1) 0.00* 0.00 0.00** 0.00**(0.00) (0.00) (0.00) (0.00)
Rain enemies of allies (t-1) 0.02 -0.07 0.04 -0.04(0.10) (0.05) (0.11) (0.04)
Sq. rain enemies of allies (t-1) 0.00 0.00 0.00 0.00(0.00) (0.00) (0.00) (0.00)
F-Stat (Kleibergen-Papp) 9.34 9.34 6.65 6.65 19.16 19.16 13.71 13.71Hansen J (p-value) 0.53 0.53 0.19 0.19 0.19 0.19 0.26 0.26Observations 1190 1190 1190 1190 1105 1105 1105 1105R-squared 0.47 0.63 0.57 0.65 0.56 0.67 0.63 0.69
Notes: An observation is a given armed group in a given year. The panel contains 85 armed groups between 1998 and 2010. Robust standard errors allowed to be clusteredat the group level in parentheses. Significance levels are indicated by * p < 0.1, ** p < 0.05, *** p < 0.01.
24
Table 4: Sample splitting.
Dependent variable: Total Fighting
(1) (2) (3) (4) (5) (6)
Tot. Fight. Enemies 0.12*** 0.10** 0.17*** 0.03 0.04 0.09*(0.04) (0.04) (0.06) (0.13) (0.04) (0.05)
Tot. Fight Allies 0.02 0.00 -0.19** -0.18*** -0.02 -0.02(0.03) (0.03) (0.08) (0.06) (0.04) (0.04)
Group FE, annual TE, rain controls Yes Yes Yes Yes Yes YesAdditional controls No Yes No Yes No YesEstimator OLS OLS IV IV IV IVSet of Instrument Variables n.a. n.a. Restricted Restricted Full FullF-Stat (Kleibergen-Papp) n.a. n.a. 6.03 4.09 100.00 26.02Hansen J (p-value) n.a. n.a. 0.38 0.08 0.55 0.09Observations 850 850 850 850 765 765R-squared 0.42 0.48 0.28 0.38 0.43 0.51
Notes: An observation is a given armed group in a given year. The panel contains 85 armed groupsbetween 1998 and 2010. Robust standard errors allowed to be clustered at the group level in parentheses.Significance levels are indicated by * p < 0.1, ** p < 0.05, *** p < 0.01.
4 Robustness Analysis
In this section, we perform a variety of robustness checks.
4.1 Alternative Specifications
In our benchmark regressions the same data (i.e., the fighting events from ACLED) are used tomeasure the network structure and the fighting effort. This is in principle not an issue, since theidentification exploits the time variation in the number of fighting events, whereas the networkstructure is time-invariant. However, it is possible to separate the two variables more sharplyby estimating the network from the earlier part of the sample, and then studying the dynamicsof the conflict using the later years of the conflict.30 More precisely, we use the information forthe subperiod 1998-2001 to estimate the network links, and the panel 2002-2010 to estimate thespillover coefficients. The results are reported in Table 4, the analogue of Table 2. The resultsare similar, albeit more fragile due to the one-third reduction in sample size. In particular, thespecification of column (3) yields similar results. However, in our preferred specification of column(6), which is more demanding, the coefficient of Total Fighting of Allies turns insignificant. It isreassuring that the signs of the 2SLS second stage coefficients are consistent in all cases with thetheoretical predictions.
The fact that the results are somewhat weaker is no surprise. On the one hand, the subperiod1998-2001 is very intense in violent events – roughly a third of the total ACLED events in oursample occurs in this subperiod. Therefore, identifying the model out of the variation over timeduring 2002-2010 is bound to yield less precise estimates. On the other hand, the network ofenmities and alliances also is measured with error, and many links are likely to be missed (we showbelow that indeed measurement error in the network links yields an attenuation bias in our model).This problem becomes more severe when we use only four years of data to estimate the links.
30It is impossible to estimate the network using data prior to 1998, since there was a major reshuffling of tacticalalliances between the First and the Second Congo War, as discussed in section 3.1.
25
In all following robustness checks (Tables 5-6), we report the results of second-stage regressionscorresponding to our preferred (benchmark) specification of column (6) in Table 2.
Table 5 considers alternative instrumentation and network construction strategies. In column(1), we use only the rainfalls in the homeland of degree 2 neighbors (i.e., the rain of enemiesof enemies and of enemies of allies) as excluded instruments, following Bramoulle, Djebbari andFortin (2009).31 The point estimates in the second-stage regression are similar to our benchmarkresults. However, they are estimated less precisely, and the coefficients of Total Fighting of Alliesturns marginally insignificant. Moreover, we face a serious weak instrument problem in the first-stage regression (where the F-stat is 3.8). This is not surprising since the degree one rainfalls arecontrolled for both in the first- and in the second-stage regressions.
In column (2) we follow a conservative identification strategy using the degree one and tworainfalls in the groups’ historical ethnic homelands as excluded instruments. By construction thesehomelands do not overlap spatially, which guarantees that the rainfalls associated with differentgroups do not overlap either. As a first step we link as many armed groups as possible to acorresponding underlying main ethnic group. This is typically the ethnic affiliation of most fighters,or at least of their leadership circle. For example, the Lord’s Resistance Army is linked to the Acholiethnic group. We find an unambiguous match for 94% of all armed groups, and drop the remaining6%. As a second step, we compute the rainfall averages on the polygons of all ethnic groups,using the digitalized version by Nunn and Wantchekon (2011) of the map of historical ethnic grouphomelands from Murdock (1959). Given that rainfall in ethnic homelands is an imperfect proxy forrainfall observed by groups in their actual current territory, the power of the first stage falls, andwe suffer from a weak instrument problem. Yet, the point estimates of the second-stage regressionare remarkably similar to the benchmark 2SLS results; only the statistical significance of TotalFighting of Allies drops.
In column (3)-(4) we address the concern that the results may be driven by many small groupsthat have a limited role in the overall conflict. In column (3) we include only groups that have atleast one enemy. This results in a sizeable reduction in the sample size. The coefficients are onaverage slightly larger in absolute value than in the baseline regression, and are both significant atthe 5% level. The F-stats of the first stage are about the threshold of 10. In column (4) we restrictthe regressions to groups that have at least one enemy and at least one ally, in order to focus on themost salient armed forces. Reassuringly, the results are quantitatively larger (again, in absolutevalue) than in the benchmark specification, and both coefficients continue to be highly significant.In summary, these robustness checks show that restricting attention to large players yields largerand more precisely estimated coefficients.
In column (5), we exclude all events involving group i when computing the total fighting effortsof allies and enemies of group i. If, for example, the enemies of the LRA are involved in 100fighting events in year 2000, out of which 30 involve the LRA, then the measure of Total Fightingof Enemies used in the regression would take the value of 70. The results are similar, although thecoefficient of Total Fighting of Enemies turns marginally insignificant.
In column (6), we also control for the total fighting of neutral groups, i.e., those classified neitheras allies nor as enemies. Our theory predicts that the coefficient of neutrals should be zero. Inline with the theory, the coefficients of Total Fighting of Enemies and Total Fighting of Allies arevirtually unchanged, and the coefficient of Total Fighting of Neutrals is zero, as predicted by thetheory.
31Note that, contrary to their model, in our theory there is no reason why an instrumentation based on first-orderlinks should yield inconsistent estimates. As discussed above, the case for our regressions to be contaminated bycontextual effects is weak in our panel regression, since time-invariant contextual effects are differenced out.
26
Finally, in column (7) we control for the lagged total fighting effort of both enemies and allied.The coefficient on the lagged variables are insignificant, whereas the coefficients of interest arehardly changed.
In Table 6, we perform a battery of additional robustness checks where we either restrict theset of fighting events considered (columns (1)-(3)), or we use alternative coding of enmities andalliances in the construction of the network (columns (4)-(7)).
The motivation for the robustness checks in columns (1)-(2) is that ideally fighting intensityshould weight the importance of the events in which each group is involved. However, detailedinformation about the size of each event (e.g., the number of casualties) is far too sparse for usto be able to run weighted regressions. Nevertheless, we have access to some binary coding fromACLED. In column (1) we drop all events with a number fatalities equal to zero (i.e. we retainthe events associated with a positive number of fatalities or for which the information about thenumber of fatalities is missing). The coefficients of interest remain very similar to the ones of thebaseline regression of column (6) of Table 2, and both variables are significant at the 5% level. Incolumn (2) we use only events that are classified in ACLED as battles, ignoring other events. Theestimated coefficients have the same sign as in the benchmark case, and are quantitatively similar.Interestingly, the estimated externalities are stronger in column (2), indicating that restrictingattention to battles, which are more important events than small riots, increases the point estimates.
In column (3) we restrict the sample to 1998-2007. The motivation is that after 2007 a fewmilitias were either annihilated (e.g., the Allied Democratic Front) or absorbed by larger combatantgroups (most notably, the DRC Army). In spite of the limited number of such episodes, thedisappearances or mergers of some groups contradicts the assumption that the network is time-invariant. Reassuringly, the estimates are of similar magnitude than in the baseline regression ofcolumn (6) of Table 2 and are statistically significant.
In column (4) we code as enemies (instead of neutrals) groups that normally fight againsteach other, but that were observed occasionally fighting on opposite sides. The magnitude ofthe coefficients remains similar, although Total Fighting of Enemies now falls short of the 10%significance threshold. In column (5) we code a dyad of armed groups as enemies if they fighteach other at least once and never fight together on the same side (recall that in the benchmarkwe required two groups to fight each other twice or more). The coefficients of the two variablesremain of similar magnitude and are highly significant. In contrast, in column (6) we follow amore restrictive rule: we code two groups as enemies if they are observed fighting on opposite sidesin at least two events (but never on the same side), and code two groups as allies if they havebeen observed fighting together on the same side at least twice (but never against each other).The two variables of interest continue to have the expected sign and are statistically significantlydifferent from zero. Finally, in column (7) we add also links among groups listed in Cunninghamet al. (2013). Our results are robust with both coefficients of interest being significant and of theexpected sign.
In Table 7 we perform a series of robustness checks with respect to the definition of groups. Onemight worry that the assumption that groups act non-cooperatively is inappropriate for a numberof strong alliances that should be better treated as unitary coalitions. In our benchmark analysiswe have followed the rule of treating groups as separate entities whenever they are classified as suchby ACLED. This agnostic way of proceeding has the advantage of not requiring any discretionalcoding decision.32 However, it is useful to check the robustness of our results in this dimension.Thus, in column (1) of Table 7 we treat all different fractions of the Rally of Congolese Democracy
32Further, note that even among strongly allied groups there is sometimes in-fighting. For example, the clashesbetween RCD’s Goma and RCD’s Kisangani fraction are infamous, as are the fights between different Mayi-Maymilitias.
27
Table 5: Robustness to alternative IVs and network construction.
Dependent variable: Total Fighting
(1) (2) (3) (4) (5) (6) (7)Degree 2 only Ethnic IV With at least With at least Excluding With neutrals With lags
1 enemy 1 ally & 1 en. bilat. evts.
Tot. Fight. Enemies 0.12** 0.08** 0.09** 0.11** 0.05 0.11** 0.08**(0.06) (0.04) (0.04) (0.05) (0.04) (0.05) (0.04)
Tot. Fight Allies -0.26 -0.15 -0.18** -0.26** -0.14** -0.16* -0.14**(0.18) (0.13) (0.09) (0.13) (0.06) (0.08) (0.07)
Tot. Fight Neutrals 0.00(0.01)
Tot. Fight. Enemies (t-1) 0.00(0.02)
Tot. Fight Allies (t-1) 0.03(0.02)
Group FE, annual TE, add. cont. Yes Yes Yes Yes Yes Yes YesZones used for IV construction Group pr. Eth. home Group pr. Group pr. Group pr. Group pr. Group pr.Excluded instruments used Deg 2 Deg 1&2 Deg 1&2 Deg 1&2 Deg 1&2 Deg 1&2 Deg 1&2
Lag 0&1 Lag 0&1 Lag 0&1 Lag 0&1 Lag 0&1 Lag 0&1 Lag 0&1F-Stat (Kleibergen-Papp) 3.82 3.55 9.95 7.87 12.03 15.72 24.73Hansen J (p-value) 0.74 0.16 0.34 0.62 0.23 0.29 0.54Observations 819 1027 819 559 1105 1105 1020R-squared 0.28 0.36 0.34 0.27 0.35 0.33 0.39
Notes: An observation is a given armed group in a given year. The panel contains 85 armed groups between 1998 and 2010. Robust standarderrors allowed to be clustered at the group level in parentheses. Significance levels are indicated by * p < 0.1, ** p < 0.05, *** p < 0.01.
28
Table 6: Robustness to alternative definitions of events and links.
Dependent variable: Total Fighting
(1) (2) (3) (4) (5) (6) (7)Only violent Only battles Only until Code incons. Code as enemy when Code as ally only when Add links from
events year 2007 as enemy opp. in at least 1 event at least 2 joint events Cunningham et al.
Tot. Fight. Enemies 0.09** 0.11** 0.09** 0.06 0.11*** 0.11** 0.10*(0.05) (0.05) (0.04) (0.05) (0.03) (0.04) (0.06)
Tot. Fight Allies -0.14** -0.17** -0.13* -0.10** -0.19** -0.10* -0.12*(0.07) (0.08) (0.08) (0.05) (0.09) (0.05) (0.07)
Group FE, annual TE, add cont. Yes Yes Yes Yes Yes Yes YesF-Stat (Kleibergen-Papp) 13.71 12.61 9.10 10.76 7.58 19.05 16.04Hansen J (p-value) 0.26 0.45 0.77 0.51 0.89 0.34 0.30Observations 1105 1066 850 1105 1105 1105 1105R-squared 0.36 0.34 0.40 0.43 0.27 0.40 0.38
Notes: An observation is a given armed group in a given year. The panel contains 85 armed groups between 1998 and 2010. Robust standard errors allowed to be clusteredat the group level in parentheses. Significance levels are indicated by * p < 0.1, ** p < 0.05, *** p < 0.01.
29
(RCD) as one single actor. Our variables of interest still have the expected sign and are significantat the 5% level. In column (2) we merge the RCD’s Goma fraction (RCD-G) with its main ally,Rwanda, and similarly we merge RCD’s Kisangani fraction (RCD-K) with its main ally, Uganda.In column (3) we merge the various DRC army fractions into two main actors. Further, in column(4) we merge all local Mayi-Mayi militias into one single actor. In column (5) all Rwandan militaryfractions are treated as a single group. Finally, in column (6) we treat as separate actors all mutiniesof the DRC army. The results are robust across all specifications.
4.2 Measurement Error in Rainfall
A concern with our IV strategy is that the rainfall variable may be subject to non-classical mea-surement error. In particular, fighting activities may destroy rain gauges located in battlefields. Asa result, our gauge-based GPCC measure might systematically underreport precipitations in warzones. The issue is twofold: first, mis-measurement may result in a spurious negative correlationbetween rainfall and fighting in the first-stage regression.33 Second, our identification hinges onrainfall in the homelands of group i′s enemies/allies having no direct effect on group i′s fightingeffort after conditioning on the rainfall in group i′s homelands. However, the exclusion restrictionwould be invalidated if the measurement error in the instruments were correlated with group i′sfighting effort.
To study this potential problem, we consider satellite rainfall estimates from TRMM or GPCP(see the data description in Section 3.2). Clearly satellite-based measurements are not affected bythe dynamics of conflict. However, they provide less direct and far less accurate rainfall estimatesthan do gauges.34 Therefore, it is not surprising that, if we use satellite rainfall data instead ofgauge-based data as instruments, we run into a severe weak instrument problem. However, thesatellite estimates can be used to infer whether gauged-based measures are biased. To this aim,consider the following simple model:
rainsat
ct = ψsat
c + rainct + vsatct (19)
raingau
ct = ψgau
c + rainct + vgauct (20)
where c denotes the grid cell at which rainfall is measured, rainct is the true (unobservable)rainfall, and vsatct and vgauct are the measurement errors. vsatct is assumed to be i.i.d.. The error termof the gauge measure is potentially subject to violence-driven measurement error. This possibilityis allowed by letting vgauct = ξ × violencect + vgauct where vgauct is an i.i.d error term. One caneliminate rainct from the previous system of equations and obtain:
raingau
ct = ψc + rainsat
ct + ξ × violencect + νct (21)
where ψc = ψgau
c − ψsat
c and νct = vgauct − vsatct are, respectively, a grid-cell fixed effect and an i.i.d.disturbance. Our null hypothesis is that ξ = 0. If ξ 6= 0, the gauge-based measure suffers withnon-classical measurement error.
We run a regression based on equation (21), measuring violence by the number of conflicts inACLED. Table 8 summarizes the results. Columns (1)-(4) report the results when satellite-basedrainfall measures are retrieved from TRMM. Column (1) is a cross-sectional specification; Column
33Remember, though, that we also use lagged rain to predict current fighting intensity.34Romilly and Gebremichael (2011) discuss the shortcomings of satellite-based rainfall estimates. On the one hand,
satellite rainfall estimates are contaminated by sources such as temporal sampling, instrument and algorithm error.On the other hand, a number of studies based on U.S. data document that their performance varies systematicallywith season, region and elevation, resulting in potentially severe biases.
30
Table 7: Robustness to alternative group definitions.
Dependent variable: Total Fighting
(1) (2) (3) (4) (5) (6)Merging all Merging RCD-G Merging all DRC Merging all Mayi- Merge Rwandan Treating mutiny
RCD fractions to Rwanda and government groups Mayi local militias military fractions events asinto a single actor RCD-K to Uganda into 2 main fractions into a single actor into a single actor separate actors
Tot. Fight. Enemies 0.12** 0.08* 0.07* 0.10** 0.09** 0.10**(0.05) (0.05) (0.04) (0.05) (0.04) (0.05)
Tot. Fight Allies -0.18** -0.10* -0.12** -0.18* -0.12** -0.14*(0.08) (0.06) (0.06) (0.10) (0.06) (0.07)
Group FE, annual TE, add cont. Yes Yes Yes Yes Yes Yes
F-Stat (Kleibergen-Papp) 13.86 16.21 11.07 8.44 15.58 14.21Hansen J (p-value) 0.25 0.50 0.37 0.64 0.27 0.54Observations 1053 1066 1079 1014 1092 1144R-squared 0.38 0.49 0.37 0.35 0.42 0.36
Notes: An observation is a given armed group in a given year. The panel contains 85 armed groups between 1998 and 2010. Robust standard errors allowed to beclustered at the group level in parentheses. Significance levels are indicated by * p < 0.1, ** p < 0.05, *** p < 0.01.
31
(2) includes grid-cell fixed effects – consistent with equation (21). In Columns (3) and (4) weconsider a log-linear specification where the two rainfall measures are log-scaled; this correspondsto a multiplicatively separable specification of model 20. Finally, we replicate the same set of fourspecifications in Columns (5)-(8) with the GPCP satellite measure. Year dummies are included inall regressions. Standard errors are clustered at the grid-cell level.35
As expected, there is a highly significant positive correlation between the gauge- and thesatellite-based rainfall measures. Most important, all estimates of ξ are not significantly differ-ent from zero, with its point estimates switching sign across specifications. The hypothesis thatξ is negative due to the destruction of gauges in battlefields is strongly rejected, especially inspecifications with grid-cell fixed effects, which are consistent with our panel specification whereparameters are identified out of the variation over time in rainfall. In these columns, the pointestimates of ξ are consistently positive and statistically insignificant. We conclude that there is noevidence that the gauge-based GPCC precipitation data are subject to non-classical measurementerror in the DRC.
4.3 Measurement Error in Network Links
Another concern here is that the network may be measured with error. Recent research by Chan-drasekhar and Lewis (2011) shows that regression of economic outcomes on network neighbors’outcomes, in the presence of measurement error of network links, can give rise to inconsistent es-timates.36 Moreover, the bias can work in different directions, and there is no general remedy tocorrect it.
In this section, we follow aMonte Carlo approach based on rewiring links in the observed networkat random, and measuring the robustness of our estimates in such perturbed networks. We considerdifferent assumptions about the extent and nature of measurement error of the network. Morespecifically, we postulate a data generating process, and then we introduce a specific (plausible)model of mis-measurement of network links. Then, we estimate the model as if the econometriciandid not know the true network, but had to infer it from data measured with error. This procedureis generated for a large number of realizations of mis-measurement errors (1,000 draws per eachcase). For each Monte Carlo draw, we consider a data generating process based on the reducedform equation (14): The true parameters correspond to our baseline estimates, β = 0.14 andγ = 0.09, and the true network corresponds to the one used in our baseline estimation afterrewiring a (sub-)set of alliances/enmities/neutral links according to a binomial process with aprobability that is invariant across the draws. Then the benchmark 2SLS specification (Column6, Table 2) is estimated on this fake dataset under the assumption that the network observed bythe econometrician is identical to the baseline one before rewiring. Iterating over the Monte Carlodraws yields the sampling distribution of the estimates of β and γ subject to link measurementerror.
The results are reported in Table 9. Each pair of columns refers to a particular mis-measurementprobability, namely the probability governing the binomial re-wiring process. From left to right, weassume that the probability that links are measured with error is zero (no measurement error), 1%,10%, 20%, 50% and one (pure noise), respectively. In all cases we report the mean and standarddeviations of the sampling distribution based on 1,000 simulations.
35Recall that the GPCP satellite measure is only available at the 2.5×2.5 degree level, i.e., for larger cells than thetwo other measures that are at the 0.5×0.5 degree level. In this case, we cluster at the 2.5×2.5 cell level.
36It has been proven, however, that likelihood-based inference while ignoring the missing data mechanism leads tounbiased estimates under the assumption of missingness at random (MAR) (Little and Rubin, 2002). Mohan et al.,(2013) provide conditions on the network for recoverability of parameters even when MAR is violated.
32
Table 8: Measurement error in rainfall.
Dependent variable: GPCC gauge rainfall measure
(1) (2) (3) (4) (5) (6) (7) (8)model linear log-linear linear log-linear
# ACLED conflict events 0.017 0.008 0.009 0.001 -0.069 0.016 -0.016 0.005(0.032) (0.012) (0.008) (0.003) (0.057) (0.014) (0.014) (0.004)
TRMM satellite rainfall measure 0.639*** 0.513*** 0.714*** 0.619***(0.018) (0.012) (0.015) (0.013)
GPCP satellite rainfall measure 0.790*** 1.073*** 0.843*** 1.233***(0.044) (0.081) (0.055) (0.100)
(0.5 x 0.5) Grid Cell FE No Yes No Yes No Yes No YesAnnual TE Yes Yes Yes Yes Yes Yes Yes Yes
Observations 9893 9893 9893 9893 9893 9893 9893 9893R-squared 0.578 0.667 0.604 0.684 0.555 0.601 0.541 0.587
Notes: An observation is a given cell of resolution 0.5 x 0.5 degrees in a given year. The panel contains 761 cells covering DRC between1998 and 2010. In Columns 3,4,5,6 all rainfall variables are in log. Robust standard errors are clustered at the (0.5 x 0.5) cell level inColumns 1-4 and at the (2.5 x 2.5) cell level in Columns 5-8. Significance levels are indicated by * p < 0.1, ** p < 0.05, *** p < 0.01.
33
In the absence of measurement error – Columns 1 and 2 – we see that the mean of the samplingdistribution is equal to the true values of the parameters. This confirms that our estimator isconsistent. Consider, next, the pair of columns corresponding to interior probabilities of mis-measurement (1%, 10%, 20% and 50%). For instance, focus on the 10% measurement error case(fifth and sixth columns). In the first row (enmity links only), we draw from a distribution whereeach enmity link has a probability 10% to be in reality a neutral relationship (no link), whereaseach neutral (no link) has a probability 10% to be in fact an enmity link. Since the number ofneutrals exceeds by far that of enmities, this perturbation implies that in reality there are moreenmity links than we observe. As expected, this experiment affects the estimate of γ more than itaffects that of β. The estimate of β is close to the no measurement error benchmark (first column)at the three digit level, whereas the estimate of γ falls from 0.091 to 0.074. Namely, measurementerror implies an attenuation bias. Next, consider the second row (alliance links only). This is thepolar opposite scenario: alliance (rather than enmity) links are measured with error. Now, theestimate of β falls from 0.141 to 0.123, while the estimate of γ increases slightly. Finally, in thethird row, we allow mis-measurement of both enmity and alliance links. In this case the estimateof β falls to 0.126 and that of γ falls to 0.08. A similar pattern is observed in the other interiorcolumns. As one moves to the right, i.e., towards larger measurement errors, the attenuation biasbecomes stronger. In the last two columns, when the econometrician observes pure noise for enmity(alliance) links, the estimate of the γ (β) tends to zero.
The lesson from this section is twofold. First, the Monte Carlo generated measurement errorsin the links leads to an attenuation bias. This suggests that, under the plausible assumption thatsome information about existing links is missing, our regression analysis underestimates the spillovereffects. Second, the extent of the bias is quantitatively modest. A measurement error of the orderof 10% (which we regard as fairly large) yields an underestimate of the spillover parameters of theorder of 11%-12%. Overall, this section confirms that our baseline estimates are robust to linkmeasurement errors.
5 Policy Analysis
In this section, we perform some policy analysis. In particular, we perform two counterfactualexercises. First, we assess the contribution of each armed group to the conflict by performing akey player analysis as outlined in Section 2.4. The analysis has important policy implications.For instance, an international organization that aims at scaling down violence may be interestedin the extent to which each of the combatant groups contributes to the conflict escalation. Anaıve approach would be to target the groups involved in the largest number of fighting episodes.However, this would ignore the endogenous network effects that may be very important. Removinga group affects in a complicated way the incentives for the surviving groups to fight. For instance, ifall agents had only first-order links, removing a group with many allies would not be useful, since itsallies would increase their effort and offset the effect of the removal of the group. To the opposite,removing a group with many enemies would yield a large reduction in violence. Unfortunately, thecalculation is far more complicated in the presence of a rich network structure, since the response ofthe surviving players depends also on higher-order links. Thus, the information about the structureof the network and the externality parameters is essential to guide the policy intervention.
The key player analysis is subject to some caveat. First, it is not clear how the internationalcommunity could wipe out large armed groups unless at the cost of an unlikely large-scale militaryoperation. Second, the removal of a large group may induce a reshuffling of alliances that our modelis not well-suited to predict. With this in mind, in the second part of this section we study the effect
34
Table 9: Monte Carlo simulations testing link mismeasurement.
Probability of mismeasurement 0 0.01 0.1 0.2 0.5 1
β γ β γ β γ β γ β γ β γ
Enmity links only Mean 0.141 0.091 0.141 0.089 0.141 0.074 0.140 0.061 0.144 0.027 0.143 0.001S.D. 0.001 0.002 0.003 0.007 0.006 0.017 0.009 0.026 0.012 0.032 0.007 0.019
Alliance links only Mean 0.141 0.091 0.138 0.092 0.123 0.096 0.103 0.097 0.044 0.098 0.001 0.091S.D. 0.001 0.001 0.004 0.013 0.016 0.025 0.025 0.028 0.036 0.032 0.009 0.010
Alliance & Enmity links Mean 0.141 0.091 0.139 0.091 0.126 0.080 0.106 0.065 0.044 0.023 0.001 -0.001S.D. 0.001 0.002 0.005 0.015 0.018 0.028 0.029 0.035 0.042 0.043 0.017 0.021
Notes: This table reports the Mean and Standard Deviations of the Monte Carlo sampling distributions (1000 draws) of the 2SLS estimates of β andγ for different probabilities of link mismeasurement. The data generating process is based on true β = 0.14 and true γ = 0.09.
35
of pacification policies, where no armed group is removed but enmity links are selectively turnedinto neutral ones. We first consider an international peace process that simultaneously mutes allenmity links in DRC. Then, we analyze and rank (less ambitious) policies that pacify armed groupsone by one.
5.1 Computation of the Counterfactual Equilibrium
Our analysis below requires us to compute counterfactual Nash equilibria corresponding to eitherthe sequential removal of each fighting group (key player) or the rewiring of some enmity links(pacification policy).
We denote by Gb the benchmark network in which all groups fight. We set the externalityparameters equal to β = 0.1407 and γ = 0.0903, in line with the point estimates of column (6) inTable 2.37 We normalize V to unity. This entails no loss of generality, since changing V wouldsimply rescale all players’ payoffs and welfare. The equations (6) and (15)–(17) allow us to estimateei, the time-invariant unobserved heterogeneity. More formally:
ei = FEi − Λβ,γ(G)(1− Λβ,γ(G)
)Γβ,γi (G), (22)
where Γβ,γi (G) = 1/(1 + βd+i − γd−i ) and Λβ,γ(G) = 1 − 1/(
∑j Γ
β,γj (G)). Consider, next, the
vector of time-varying shifters zit (rainfall, etc.). We perform the key player analysis in an averagescenario in which all time-varying group-specific shifters are collapsed to their sample average,zi =
∑2010t=1998
zit13 , and denote by Z = {zi} the estimated matrix of shifters. In other words, we
compare an average year of conflict in the benchmark model to its corresponding counterfactual.Finally, we set the average of the time-varying i.i.d. unobserved shifter,
∑2010t=1998 ǫit, equal to zero
for all groups.We can then compute the Nash equilibrium. Following the analysis in Section 2.6, the vector
of equilibrium fighting efforts is obtained by inverting the system of equilibrium conditions impliedby equations (15) and (16). In matrix form, this yields:
x∗(Gb) = (I+ βA+(Gb)− γA−(Gb))−1[Λβ,γ(Gb)(1− Λβ,γ(Gb))Γβ,γ(Gb)− (Zα+ e)
](23)
5.2 Key Player Analysis
To perform the key player analysis, we follow the procedure outlined above for each counterfactualequilibrium Gb\{k} (where Gb\{k} denotes the network after the removal of group k). The vectorof equilibrium fighting efforts is given by equations which are analogous to equation (23) exceptthat the dimension of the system is reduced by one, the adjacency matrix is A(Gb\{k}) and the
parameters attached to the network structure are replaced by Λβ,γ(Gb\{k}) and Γβ,γ(Gb\{k}). Wecompute rent dissipation before and after the removal of an agent k. Then, the change in the
rent dissipation, which can be interpreted as the reduction in aggregate fighting, equals ∆RDβ,γk ≡
RDβ,γ(Gb)− RDβ,γ
(Gb\{k}
), where RDβ,γ
(Gb)≡∑n
i=1 x∗i
(Gb).
Figure 7 plots the percentage change in the rent dissipation, ∆RDβ,γk versus the observed share
in total fighting x∗k(Gb)/∑n
i=1 x∗i
(Gb)focusing on the 20 groups whose removal yields the largest
37All second-order conditions (cf. equation (3)) continue to hold for all groups in the counterfactual experimentsin which one player is removed.
36
reduction in rent dissipation.38 The Figure is in a log scale to ease the visualization of the mainactors (red acronyms denote groups affiliated to foreign governments). Appendix Table 1 reports
a complete ranking of the agents k according to the change in the rent dissipation ∆RDβ,γk . The
table also reports the contribution of each player to the total fighting when all groups are active.Two findings are noteworthy. First, although the observed contribution of each group to total
fighting correlates significantly with the reduction in total fighting associated with its removal,it is significantly below one.39 For instance, the Rwanda-backed Rally for Congolese Democracy(Goma) is the single most active armed group in the DRC war (excluding the DRC government)accounting for ca. 8.6% of the total military activity in the DRC. However, its removal wouldreduce aggregate fighting by only 2.3%. In contrast, removing the FDLR, the main Hutu rebelgroup that fiercely opposes Tutsi influence in the region, would yield a reduction in violence of13.8%, despite the fact that this group is responsible for less than 7% of the fighting activity.Some notorious (and virulent) rebel groups such as the LRA (Lord Resistance Army) and LaurentNkunda’s CNDP (National Congress for the Defense of People) are not among the top 20. Nor isthe RCD-Kisangani, in spite of being the second most active group.
Second, foreign armies rank among the most disruptive players in the conflict. These include,in a ranked order, the armies of Rwanda, Uganda, Zimbabwe, Angola, South Africa, Sudan andBurundi (with Zambia and Chad also ranking among the top 30). It is remarkable that eight out ofthe twenty most virulent groups are foreign national armies. This confirms the anecdotal evidencethat foreign intervention has been key for the escalation of the DRC war. To check the robustness ofthis result, we constructed a counterfactual in which all foreign troops are removed simultaneously.In this case, total fighting decreases by 24 percent!
Figure 8 shows the change in the rent dissipation as a function of the key player ranking. Onthe one hand, there are only 7 armed groups whose removal would reduce conflict by more than2.5%. This shows how difficult it is to obtain large results by targeting individual players in aconflict with so many interconnected armies and militias. On the other hand, there are 5 groupswhose removal would lead to a significant (2.5% or more) escalation rather than containment ofthe conflict.40
5.3 Pacification Policies
Using a similar approach, we study the effect of pacification policies aimed at reducing ethnicand political hostility across groups. As in the key player analysis, we study the change in rentdissipation associated with counterfactual scenarios. However, instead of removing some groups, weturn, selectively, some enmity links into neutral links. We start with a drastic (albeit unrealistic)counterfactual in which all enmity links are rewired into neutral relationships. The effect is verylarge: aggregate fighting is reduced by 54%. Not only enmities but also alliance links are importantfor the containment of the conflict: re-wiring all friendship links into neutral relationships yieldsincrease in aggregate fighting by 95%. Thus, an even more dramatic counterfactual scenario isobtained by rewiring all enmity links and neutral relationships into friendship (i.e., alliance) links.The result is a reduction of aggregate violence in the order of 91% - almost full peace. This result
38We exclude from this ranking and alternative ones the DRC Army, since the counterfactual removal of thenational army of the country is not an interesting scenario. The DRC army is active in both the benchmark and allcounterfactual scenarios. We simply omit it from the figures and tables below.
39The correlation is between the two variables is 71%. The corresponding Spearman rank correlation is 90%.40These groups are Mutiny of the Military Forces of DRC (-3.3%), Congolese Libration Movement (-5.2%), RCD-
Kisangani (-5.4%), LRA (-7.5%), and CNDP (-8.8%).
37
100
101
100
101
FDLR
MAYIRWA-II
UGA
LENDURWA-I
RCD
INTERAH RCD-G
UPCZBWHUTU
SAF
ANG
MAYI-PNAM
PUSICEX-RWA
SUDBUR
Share in tot. fight., x∗
k(Gb)/
∑nj=1x
∗
j(Gb) [%]
Changein
rentdiss.
removingk,∆RD
β,γ
k[%
]
1
1 5 10
5
10
Figure 7: The figure reports the relative change in the rent dissipation associated with the removalof each of the top 20 groups in the ranking of the key player analysis versus the respective sharein total fighting before removal. Both axes are in log-scale. The red dashed line indicates the 45degree line. The acronyms stand for the following: FDLR = Democratic Forces for the Liberationof Rwanda, RWA-II = Mil. Forc. of Rwanda (2000-), MAYI = Mayi-Mayi Milita, UGA = Mil.Forc. of Uganda (1986-), LENDU = Lendu Ethnic Militia (DRC), RWA-I Mil. Forc. of Rwanda(1994-1999) RCD = Rally for Congolese Dem., INTERAH = Interahamwe Hutu Ethnic Militia,RCD-G = Rally for Congolese Dem. (Goma), UPC = Union of Congolese Patriots, ZBW = Mil.Forc. of Zimbabwe (1980-), HUTU = Hutu Rebels, SAF = Mil. Forc. of South Africa (1994-1999),ANG = FAA/MPLA = Mil. Forc. of Angola (1975-), MAYI-P = Mayi Mayi Militia(PARECO),NAM= Mil. Forc. of Namibia (1990-2005), PUSIC = Party for Unity and Safekeeping of Congo’sIntegrity, EX-RWA = Former Mil. Forc. of Rwanda,(1973-1994), SUD = Mil. Forc. of Sudan(1993-) and BUR = Mil. Forc. of Burundi (1996-2005). Group names indicated in red refer togovernment groups.
38
10 20 30 40 50 60 70 80−10
−5
0
5
10
15
rank k
∆RD
β,γ
k[%
]
Figure 8: The figure shows the relative change in the rent dissipation associated with the removalof group k as a function of the key player ranking. Forces affiliated to the DRC army are notincluded in the figure.
echoes the insight of the theoretical analysis of the regular graph in Section 2.5.41
Since an intervention that wipes out all enmities in DRC would be utopistic, we consider a lessambitious targeted policy aimed to mute all enmity links associated with individual groups (i.e., wepacify one group at a time). Figure 9 shows the results (in logarithmic scale) focusing on the localmilitias whose pacification triggers a reduction in violence larger than 1%.42 The largest effect isobtained from pacifying the RCD-Goma (17%), followed by FDLR (9.68%) and RCD-Kisangani(8.74%). As the figure shows, some of these groups are small players in terms of total contributionto violence. Appendix Table 2 reports the rent dissipation for all groups. Surprisingly, there area few cases in which the pacification policy appears to be counterproductive: Ngiti Ethnic Militia,Alliance for Democratic Change, Alur Ethnic Militia.
6 Conclusion
In this paper, we construct a theory of conflict in which different groups compete over a fixed amountof resources. We introduce a network of alliances and enmities that we model as externalities addedto a Tullock contest success function. Alliances are beneficial to each member, but are not unitarycoalitions. Rather, each player acts strategically vis-a-vis both allies and enemies. We view ourtheory as especially useful in conflicts characterized by high fragmentation, non-transitive relationsand decentralized military commands, all common features of civil conflicts.
We apply the theory to the analysis of the Second Congo War, one of the bloodiest civil conflictsin modern history. Our estimation of the network externalities is similar methodologically to thatfollowed in the recent work of Acemoglu, Garcia-Jimeno and Robinson (2014) who tackle a reflectionproblem through an instrumental variable strategy. While they rely on historical information,
41The counterfactual scenarios above can be alternatively interpreted as a quantitative assessment of the mecha-nisms of our theory in explaining the data. Rewiring all alliances into neutral links is identical to setting the fightingspillover β = 0, whereas rewiring all enmities into neutral links is identical to setting γ = 0.
42The effect of pacifying foreign armies would be very large. However, the interpretation of this policy is ambiguouswhen applied to national armies. The reduction in rent dissipations associated with pacifying foreign armies are:Angola 77%, Namibia 21%, Rwanda 17%, Zimbabwe 15%, Chad 12%, Sudan 11%, Uganda 8%, South Africa 3%.
39
10−1
100
101
100
101
RCD-G
FDLRRCD-K
LRA
CNDPLENDU
MAYI-J / MAYI-F MAYIMAYI-K
Share in tot. fight., x∗
k(Gb)/
∑nj=1x
∗
j(Gb) [%]
Changein
rentdiss.
removingkj,∆RD
β,γ
kj[%
]
1 10
1
0.1
10
Figure 9: The figure reports the relative change in the rent dissipation associated with the pacifi-cation of enmity links with the DRC government of the top 10 local groups in the ranking of thekey links analysis versus the respective share in total fighting before pacification. Both axes are inlog-scale. The red dashed line indicates the 45 degrees line. The acronyms stand for the following:FDLR = Democratic Forces for the Liberation of Rwanda, MAYI = Mayi-Mayi Milita, LENDU= Lendu Ethnic Militia (DRC), LRA = Lord’s Resistance Army, RCD-G = Rally for CongoleseDem. (Goma), RCD-K = Rally for Congolese Dem. (Kinsangani), CNDP = Nat. Congress forthe Defense of the People, MAYI-K = Mayi-Mayi Militia (Kifuafua), MAYI-F = Mayi Mayi Milita(Cmdt La Fontaine) and MAYI-J = Mayi-Mayi Militia (Cmdt Jackson).
40
we exploit the exogenous variation over space and time in weather conditions. The signs of theestimated coefficients conform with the prediction of the theory. Each group’s fighting effort isincreasing in the total fighting of its enemies and decreasing in the total fighting of its allies. Wethen use our structural model to quantify the efficiency of various pacification policies. In particularwe perform a key player analysis to identify which groups contribute most to the escalation of theconflict, either directly or indirectly, via the externalities they exercise on the other groups’ fightingeffort. The analysis highlight a number of interesting results, such as the key role of foreign armiesin the conflict escalation.
The Congo War is a natural testing ground for our theory for being a conflict where mostalliances and enmities are shallow links, and where many allied actors do not coordinate theiractions. However, informal alliances and enmities and intransitive links are by no means uniqueto Congo. Rather, they are common fare in most modern civil conflicts, and pervasively so in forexample the conflicts of Afghanistan, Somalia, Iraq, Sudan and Syria.
Even in the case of more conventional international wars, shallow links and intransitive linksare not uncommon. For instance, the anti-Nazi alliance between the Soviet Union and the Anglo-Americans during World War II was a tactical alliance to defeat a common enemy. Well before thewar was over, the Soviet Union and the Anglo-Americans were fighting strategically for conflictingobjectives, each trying to secure the best political and military post-war outcome.43 Anotherexample is the intricate situation in the Balkans during WWII.44 Similar considerations apply toearlier wars, from the Peloponnesian War in ancient Greece, to the Napoleonic Wars (Ke, Konrad,and Morath 2013), or to the alliances between warlords in China after the proclamation of theRepublic in 1912.
Our analysis takes the first step towards understanding how webs of alliances and enmities canlead to escalation or containment of conflict. An important limitation is that we take the networkas exogenous, and do not try to model its formation or dynamic evolution. Endogenizing networkformation is a challenging extension that we leave to future research.
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44
Appendix
Appendix A Proofs
The proof of Proposition 1 proceeds by first deriving the necessary first order conditions (FOCs),and then establishing that the second order conditions (SOCs) hold. For the latter, we provide acondition (cf. equation (3)) on the admissible parameters such that agent i’s payoff, πi(x
∗−i, xi, G),
given the other agents’ efforts, x∗−i, is a locally concave function of xi for all agents i = 1, . . . , n.
This implies that the necessary FOCs are also sufficient and hence pin down a local maximum forall agents.
Formally, this guarantees that the strategy profile x∗ is a local Nash equilibrium, which isdefined as follows:
Definition 2 (Alos-Ferrer and Ania, 2001). Consider a game with a finite number n of players,where the strategy spaces Si are connected subsets of finite-dimensional Euclidean spaces, and thepayoff functions πi : S → R where S = S1 × · · · × Sn. A “local Nash equilibrium” for the game is astrategy profile (xi)
ni=1 ∈ S such that, for all players i, xi is a local maximum of πi (·,x−i), that is,
there exists ε > 0 such that, for all x′i in an ε-neighborhood of xi, πi (xi,x−i) ≥ πi (x′i,x−i) .
To ensure that this strategy profile x∗ is, in addition, a Nash equilibrium in the standard sense,we must impose a lower bound xi (cf. equation (36) in the proof of Proposition 1) on the strategyspace of each agent to ensure that, given the other agents’ efforts x∗
−i, for agent i, x∗i is a bestreply over the entire admissible strategy space of i, given by [xi,∞). We then show that a commonlower bound on the strategy spaces is simply defined by x ≡ maxi=1,...,n xi. Under the conditionxi ∈ [x,∞) for each agent i we also obtain that x∗ is a Nash equilibrium.
Proof of Proposition 1. For our equilibrium characterization we start with the computation ofthe FOCs. With equation (2) we can write the payoff of agent i as follows
πi(G,x) = Vϕi∑nj=1 ϕj
− xi
= Vxi + β
∑nj=1 a
+ijxj − γ
∑nj=1 a
−ijxj
∑nj=1
(xj + β
∑nk=1 a
+jkxk − γ
∑nk=1 a
−jkxk
) − xi. (24)
The partial derivatives are given by
∂πi∂xi
= V
∂ϕi∂xi
∑nj=1 ϕj − ϕi
∑nj=1
∂ϕj
∂xi(∑nj=1 ϕj
)2 − 1
= V
∑nj=1ϕj − ϕi(1 + βd+i − γd−i )(∑n
j=1 ϕj
)2 − 1, (25)
where we have used the fact that∂ϕj
∂xi= δij+βa
+ij−γa
−ij and consequently
∑nj=1
∂ϕj
∂xi= 1+βd+i −γd
−i .
The FOCs are then given by ∂πi∂xi
= 0. A simple manipulation of the FOC for each player i yieldsthe individual operational performance
ϕi =1
1 + βd+i − γd−i
1−
1
V
n∑
j=1
ϕj
n∑
j=1
ϕj .
45
Summation over i = 1, . . . , n gives the aggregate operational performance
n∑
i=1
ϕi = V
1−
1∑ni=1
11+βd+i −γd−i
.
In the following we denote by
Γβ,γi (G) ≡
1
1 + βd+i − γd−i,
Λβ,γ(G) ≡ 1−1∑n
i=11
1+βd+i −γd−i
= 1−1
∑ni=1 Γ
β,γi (G)
,
so that we can write aggregate operational performance at equilibrium as
n∑
i=1
ϕi = V Λβ,γ(G), (26)
and individual operational performance at equilibrium as
ϕi = V Λβ,γ(G)(1 − Λβ,γ(G))Γβ,γi (G). (27)
As shown below, by imposing restrictions such that the SOCs are satisfied (cf. equation (32)), weobtain as a result that that the operational performances in equation (27) are non-negative. Note
that for the local hostility levels Γβ,γi (G) = 1
1+βd+i −γd−ito be non-negative for all i = 1, . . . , n, we
need to require that 1+βd+i −γd−i ≥ 0. This holds if γd−i ≤ 1, so that we require that γ < 1/d−max.It further follows that 1 − Λβ,γ(G) = 1
∑ni=1 Γ
β,γi (G)
> 0 for a finite network (cf. Lemma 3). Next,
we have that Λβ,γ(G) = 1 − 1∑n
i=1 Γβ,γi (G)
> 0 is equivalent to∑n
i=1 Γβ,γi (G) > 1. Observe that
∑ni=1 Γ
β,γi (G) =
∑ni=1
11+βd+i −γd−i
≥∑n
i=11
1+βd+i≥ n
1+βd+max. This is greater than one if β < n−1
d+max,
and this holds if β < 1. By requiring that β + γ < 1/max{λmax(G+), λmax(G
−)} we have thatβ < 1 for any non-empty graph, and ϕi ≥ 0 for all i = 1, . . . , n in equation (27).
We next compute the equilibrium effort levels, based on the definition of operational perfor-mance (2) that we combine with equation (27). We have that
xi + βn∑
j=1
a+ijxj − γn∑
j=1
a−ijxj = V Λn,β(G)(1 − Λn,β(G))Γβ,γi (G), (28)
Denoting by Γβ,γ(G) ≡ (Γβ,γ1 (G), . . . ,Γβ,γ
n (G))⊤, we can write this in vector-matrix form as
(In + βA+ − γA−)x = V Λβ,γ(G)(1 − Λβ,γ(G))Γβ,γ(G) (29)
We now proceed by deriving sufficient conditions such that the matrix In+βA+−γA− is invertible.
Provided that the SOCs hold (see below), this also guarantees the existence and uniqueness of theNash equilibrium. Observe that the matrix In + βA+ − γA− is invertible if its determinant is notzero. We first consider two special cases. First, assume that γ = 0. Then the inverse (In + βA)−1
exists only if |In+βA| 6= 0. We have that |In+βA| = 0 if and only if − 1β is an eigenvalue ofA. If the
smallest eigenvalue λmin is such that β < 1/|λmin|, then this condition is satisfied. This is because|λmin| <
1β implies that − 1
β < λmin and we must have that − 1β 6= λi for all i = 1, . . . , n. Second,
46
assume that β = 0. Then the matrix In− γA− is invertible if γ < 1/λmax(G−), where λmax(G
−) isthe largest real eigenvalue of G−. In the general case a sufficient condition for the matrix In+βA
+−γA− to be invertible, is that β+γ < 1/max{λmax(G
+), λmax(G−)}. This is because the determinant
of a matrix of the form In −∑p
j=1 λjWj is strictly positive if∑p
j=1 |λj| < 1/maxj=1,...,p ‖Wj‖,where ‖Wj‖ is any matrix norm, including the spectral norm, which corresponds to the largesteigenvalue of Wj (cf. e.g. Lee and Liu, 2010). A strictly positive determinant then impliesinvertibility. Further, note that λmax(G
−) < d−max (Cvetkovic, Doob and Sachs 1995), so thatβ + γ < 1/max{λmax(G
+), d−max} implies that β + γ < 1/max{λmax(G+), λmax(G
−)}. From ourprevious discussion we see that this also guarantees that the operational performances in equation(27) are non-negative.
In the following we provide an explicit computation of the equilibrium fighting efforts wheninvertibility holds. More precisely, when the matrix In + βA+ − γA− is invertible, we obtain from(29) the equilibrium effort levels,
x∗ = V Λβ,γ(G)(1 − Λβ,γ(G)) (In + βA+ − γA−)−1Γβ,γ(G)︸ ︷︷ ︸cβ,γ(G)
,
where we have introduced the centrality
cβ,γ(G) ≡ (In + βA+ − γA−)−1Γβ,γ(G).
We can write the equilibrium effort for each agent i as follows
x∗i = V Λβ,γ(G)(1 − Λβ,γ(G))cβ,γi (G). (30)
This is exactly equation (8) in the proposition. Moreover, equilibrium payoff is given by
π∗i (G) ≡ πi(x∗, G) = V
ϕ∗i (G)∑n
j=1ϕ∗j(G)
− x∗i = V (1− Λβ,γ(G))(Γβ,γi (G) − Λβ,γ(G)cβ,γi (G)
),
and we obtain equation (9) in the proposition.In order to show that the necessary first order conditions are also sufficient, we next compute
the second order conditions (SOCs). From equation (25) we have that the second partial crossderivative of agent i’s payoff is given by
∂2πi∂xi∂xj
=V
(∑n
k=1 ϕk)4
(
n∑
k=1
ϕk
)2( n∑
k=1
∂ϕk
∂xj− (1 + βd+i − γd−i )
∂ϕi
∂xj
)
−2
(n∑
k=1
ϕk
)(n∑
k=1
ϕk − ϕi(1 + βd+i − γd−i )
)n∑
k=1
∂ϕk
∂xj
].
Using the fact thatn∑
k=1
∂ϕk
∂xj= 1 + βd+j − γd−j ,
and that
Γβ,γi (G) =
1
1 + βd+i − γd−i,
n∑
k=1
ϕk = V Λβ,γ(G) = V1
∑ni=1 Γ
β,γi (G)
(n∑
i=1
Γβ,γi (G)− 1
)= V
u⊤Γβ,γ(G) − 1
u⊤Γβ,γ(G),
47
we can write the second partial cross derivative as
∂2πi∂xi∂xj
=1
V Λβ,γ(G)2
[1− 2Λβ,γ(G)
Γβ,γj (G)
−δij + β1{j∈N+
i } − γ1{j∈N−i }
Γβ,γi (G)
]. (31)
From equation (31) we find that
∂2πi∂x2i
= −2
V Λβ,γ(G)Γβ,γi (G)
= −2
V
∑nj=1 Γ
β,γj (G)
∑nj=1 Γ
β,γj (G) − 1
(1 + βd+i − γd−i ), (32)
which is negative if Λβ,γ(G) > 0 and 1+βd+i − γd−i > 0. The last inequality requires that the local
hostility level of agent i is positive, i.e. Γβ,γi (G) > 0. It then follows that the equilibrium payoff
function is locally concave under the condition in equation (3).In the remainder of the proof we show that the equilibrium payoff function is globally concave
over a specified domain, by bounding the effort levels (from below). This implies that the marginalpayoff of an agent from deviating from the Nash equilibrium strategy is always negative. Observethat
πi(x∗−i, xi, G) = V
ϕi(x∗−i, x,G)∑n
j=1 ϕj(x∗−i, x,G)
− xi
= Vϕi(x
∗, G)− (x∗i − xi)∑nj=1 ϕj(x
∗, G) − (1 + βd+i − γd−i )(x∗i − xi)
− xi
= VAβ,γ
i (G) + Γβ,γi (G)xi
Bβ,γi (G) + xi
− xi, (33)
where we have denoted by
Aβ,γi (G) ≡ Γβ,γ
i (G)(ϕ∗i − x∗i ) = V Λβ,γ(G)(1 − Λβ,γ(G))Γβ,γ
i (G)(Γβ,γi (G)− cβ,γi (G)
),
Bβ,γi (G) ≡ Γβ,γ
i (G)
n∑
j=1
ϕ∗j − x∗i = V Λβ,γ(G)Γβ,γ
i (G)− x∗i = V Λβ,γ(G)(Γβ,γi (G) − (1− Λβ,γ(G))cβ,γi (G)
),
and the centrality cβ,γi (G) has been defined in equation (7). We first assume that Bβ,γi (G) >
0. Figure I illustrates the function πi(x∗−i, x,G) in equation (33) for different values of xi. For
xi > −Bβ,γi (G) it is concave with a maximum at x∗i , and decreasing for values of xi > x∗i with
an asymptote given by −xi.45 In particular, if xi is larger than the discontinuity at −Bβ,γ
i (G) ofπi(x
∗−i, x,G) then we have that πi(x
∗−i, xi, G) < πi(x
∗, G). Note that πi(x∗−i, xi, G) < πi(x
∗, G) iff
VAβ,γ
i (G) + Γβ,γi (G)xi
Bβ,γi (G) + xi
− xi < π∗i = Vϕ∗i∑n
j=1ϕ∗j(G)
− x∗i
= V (1− Λβ,γ(G))Γβ,γi (G) − x∗i
= V (1− Λβ,γ(G))(Γβ,γi (G)− Λβ,γ(G)cβ,γi (G)
), (34)
45 Note that∂2πi(x
∗
−i,xi,G)
∂x2i
= −2V
(
Bβ,γi
(G)−Aβ,γi
(G)/Γβ,γi
(G))
Γβ,γi
(G)
(Bβ,γi
(G)+xi)3
< 0 for xi > −Bβ,γi (G) and using the fact
that Bβ,γi (G) − Aβ,γ
i (G)/Γβ,γi (G) = Γβ,γ
i (G)∑
j ϕ∗j − x∗
i − ϕ∗i + x∗
i = V Γβ,γi (G)Λβ,γ(G) − ϕ∗
i = V Λβ,γ(G)Γβ,γi (G)−
V Λβ,γ(G)(1− Λβ,γ(G))Γβ,γi (G) = V Λβ,γ(G)2Γβ,γ
i (G) > 0.
48
-6 -4 -2 0 2 4 6
-10
-5
0
5
10
xi
πi(x∗ −i,xi,G)
Figure I: The payoff function πi(x∗−i, xi, G) from equation (33) for different values of xi. The
horizontal red dashed line indicates π∗i , the red vertical dashed line indicates x∗i , while the blue
vertical solid line indicates the point of discontinuity, −Bβ,γi (G). If xi is larger than the discontinuity
at −Bβ,γi (G) then we have that πi(x
∗−i, xi, G) < πi(x
∗, G). The blue dashed line indicates theasymptote with slope −1.
where we have denoted by π∗i ≡ πi(x∗, G). Next, observe from the FOC that
∂πi(x∗−i, xi, G)
∂xi= −
Aβ,γi (G)−Bβ,γ
i (G)Γβ,γi (G) + (Bβ,γ
i (G) + xi)2
(Bβ,γi (G) + xi)2
= 0, (35)
it follows that46
xi = −Bβ,γi (G) +
√Bβ,γ
i (G)Γβ,γi (G) −Aβ,γ
i (G) = x∗i ,
and inserting into πi(x∗−i, xi, G) yields π
∗i . We then define
xi ≡ −Bβ,γi (G). (36)
Assuming that Bβ,γi (G) ≥ 0 we get xi < 0. Note that if Bβ,γ
i (G) > 0 then the sign of Aβ,γi (G)
does not qualitatively change the functional form of πi(x∗−i, x,G) (more precisely, πi(x
∗−i, x,G) is a
concave function for xi ≥ −Bβ,γi (G) with a global maximum over its domain at x∗i ; see also footnote
45). In contrast, if Bβ,γi (G) < 0 then we must have that Aβ,γ
i (G) must be negative as well. More
precisely, if Bβ,γi (G) < 0 then we must have that Aβ,γ
i (G) < Bβ,γi (G)Γβ,γ
i (G). Otherwise we would
have that∂πi(x∗
−i,xi,G)
∂xi< 0 (see equation (35)), violating the FOC. However, under this condition,
the functional form of πi(x∗−i, x,G) remains qualitatively unchanged.
To summarize, for all the above cases we have that πi(x∗−i, xi, G)−πi(x
∗, G) < 0 if xi ∈ [xi,∞).Hence, this characterizes a Nash equilibrium. This completes the proof.
Remark 1. It is possible to compute a common lower bound in Proposition 1, x, on the effort levelsxi, such that πi(x
∗−i, xi, G) − πi(x
∗, G) < 0 holds for all i = 1, . . . , n if x ∈ [x,∞)n. Moreover,
46W.l.o.g. set V = 1. Then, using the fact that Aβ,γi (G) = Γβ,γ
i (G)Λβ,γ(G)(1 − Λβ,γ(G))(Γβ,γi (G) − cβ,γi (G)),
Bβ,γi (G) = Λβ,γ(G)(Γβ,γ
i (G)−(1−Λβ,γ(G))cβ,γi (G)) and x∗i = Λβ,γ(G)(1−Λβ,γ(G))cβ,γ
i (G), we find that −Bβ,γi (G)+
√
Bβ,γi (G)Γβ,γ
i (G)−Aβ,γi (G) = Λβ,γ(G)(1− Λβ,γ(G))cβ,γi (G) = x∗
i .
49
we also require that x < x∗i for all i = 1, . . . , n. We denote this common lower bound as x ≡maxi=1,...,n xi, and we require that ∀i = 1, . . . , n
x ≤ x∗i = V Λβ,γ(G)(1 − Λβ,γ(G))cβ,γi (G). (37)
Note that equation (37) is equivalent to
maxi=1,...,n
{−Bβ,γ
i (G)}= max
i=1,...,n
x
∗i − Γβ,γ
i (G)
n∑
j=1
ϕ∗j
≤ min
i=1,...,nx∗i ,
or
maxi=1,...,n
{cβ,γi (G)−
Γβ,γi (G)
1− Λβ,γ(G)
}≤ min
i=1,...,ncβ,γi (G),
Note that this is equivalent to
maxi=1,...,n
Γβ,γi (G)︸ ︷︷ ︸
≥0 by Prop. 1
cβ,γi (G)/Γβ,γ
i (G)−n∑
j=1
Γβ,γj (G)
︸ ︷︷ ︸≤0 if Bβ,γ
i (G)≥0
≤ mini=1,...,n
cβ,γi (G) (38)
Proof of Lemma 1. The proof of the lemma builds on a first order Taylor approximation in βand γ of the centrality cβ,γ(G) defined in equation (7). Using the fact that In+βA
+−γA− = (In+βA+)(In−γA
−)+βγA+A− = (In+βA+)(In−γA
−)+O(βγ), and [(In + βA+)(In − γA−)]−1
=(In−γA
−)−1(In+βA+)−1, where (In−γA
−)−1 =∑∞
k=0 γk(A−)k and (In+βA
+)−1 =∑∞
k=0(−1)kβk(A+)k,we can write
(In + βA+ − γA−)−1 = (In − γA−)−1(In + βA+)−1 +O(βγ)
=
(In + γA− +
∞∑
k=2
γk(A−)k
)(In − βA+ +
∞∑
k=2
(−1)kβk(A+)k
)+O(βγ)
= In + γA− − βA+ +∞∑
k=2
γk(A−)k +∞∑
k=2
(−1)kβk(A+)k + γA−∞∑
k=2
(−1)kβk(A+)k
+
∞∑
k=2
γk(A−)kβA+ +
(∞∑
k=2
γk(A−)k
)(∞∑
k=2
(−1)kβk(A+)k
)+O(βγ)
= In +
∞∑
k=1
γk(A−)k +
∞∑
k=1
(−1)kβk(A+)k +O(βγ). (39)
Hence, in leading order in β and γ the centrality in equation (7) can then be written as follows
cβ,γ(G) =
(∞∑
k=0
γk(A−)k +∞∑
k=0
(−1)kβk(A+)k − In
)Γβ,γ(G) +O(βγ)
= bΓβ,γ(G)(γ,G−) + bΓβ,γ(G)(−β,G
+)− Γβ,γ(G) +O(βγ).
This completes the proof.
50
0.0 0.1 0.2 0.3 0.4 0.5
3
4
5
6
7
β
∑n i=
1cβ
,γi
(G)
γ = 0.01
γ = 0.05
γ = 0.1
Figure II: (Left panel) Illustration of a bowtie graph, where the central agent is in conflict with allother agents and the peripheral pairs of agents are allied. Alliances are indicated with thick lineswhile conflict relationships are indicated with thin lines. (Right panel) The first-order approxima-tion (dashed lines) used to derive the equilibrium efforts in equation (40), and the exact value (solid
lines) for the total centrality∑n
i=1 cβ,γi (G) for different values of β and γ in the bowtie graph.
Proof of Lemma 2. The proof of the lemma builds on a first order Taylor approximation in βand γ of the equilibrium fighting efforts x∗i (G) in equation (8) and payoffs π∗i (G) from equation(9). First, observe that for β → 0 and γ → 0 we have that
Γβ,γi (G) =
1
1 + βd+i − γd−i= 1− βd+i + γd−i +O
(β2)+O
(γ2)+O (βγ) ,
and
1− Λβ,γ(G) =1
∑ni=1 Γ
β,γi (G)
=1
n+
2
n2(βm+ − γm−
)+O
(β2)+O
(γ2)+O (βγ) ,
where we have denoted by m+ = 12
∑ni=1 d
+i and m− = 1
2
∑ni=1 d
−i . Also, we have that
Λβ,γ(G)(1 − Λβ,γ(G)) =n− 1
n2−
2m−(n− 2)γ
n3+
2m+(n− 2)β
n3+O
(β2)+O
(γ2)+O (βγ) .
Moreover, from equation (39) in the proof of Lemma 1 we have that
(In + βA+ − γA−)−1 = (In − γA−)−1(In + βA+)−1 +O (βγ)
= (In + γA−)(In − βA+)(u+ γd+ − βd− +O(β2)+O
(γ2)+O (βγ)
= In + γA− − βA+ +O(β2)+O
(γ2)+O (βγ) .
It then follows that
(In + βA+ − γA−)−1Γβ,γ(G) = (In + γA− − βA+)(u+ γd− − βd+) +O(β2)+O
(γ2)+O (βγ)
= u+ 2γd− − 2βd+ +O(β2)+O
(γ2)+O (βγ) ,
and we get for the equilibrium fighting efforts x∗i (G) that
x∗i (G) = V Λβ,γ(G)(1 − Λβ,γ(G))((In + βA+ − γA−)−1Γβ,γ(G))i
= V
(n− 1
n2−
2m−(n− 2)γ
n3+
2m+(n− 2)β
n3
)(1 + 2γd−i − 2βd+i ) +O (βγ)
= V
(n− 1
n2−
2(n− 2)m− − 2n(n− 1)d−in3
γ +2(n − 2)m+ − 2n(n− 1)d+i
n3β
)+O (βγ) .
51
Denoting by Aβ,γ(G) ≡ n−1n2 +β 2(n−2)
n3 m+−γ 2(n−2)n3 m− and B ≡ 2(n−1)
n2 , we can write this as follows
x∗i (G) = V(Aβ,γ(G)−B
(βd+i − γd−i
))+O (βγ) . (40)
Next, note that the payoff of agent i in equilibrium is given by π∗i (G) = V (1− Λβ,γ(G))Γβ,γi (G)−
x∗i (G). Using the equilibrium effort from above it then follows that the equilibrium payoff of agenti is given by
π∗i (G) = V (1− Λβ,γ(G))Γβ,γi (G) − x∗i (G)
=
(1
n−
2γm− − 2βm+
n2
)(1 + γd−i − βd+i )− x∗i (G) +O (βγ)
= V
(1
n2+
4m+ + n(n− 2)d+in3
β −4m− + n(n− 2)d−i
n3γ
)+O (βγ) .
Denoting by Cβ,γ(G) ≡ 1n2 + β 4
n3m+ − γ 4
n3m− and D ≡ n−2
n2 , this can be written as follows
π∗i (G) = V(Cβ,γ(G) +D
(βd+i − γd−i
))+O (βγ) .
This completes the proof.
Figure II shows an illustration of the first-order approximation used to derive the equilibriumefforts in equation (40), and the exact value for the total centrality
∑ni=1 c
β,γi (G) for different values
of β and γ for a bowtie graph.
52
Online Appendix B Proofs and Technical Analysis
In the following lemma we state additional results on the monotonicity and bounds on the localhostility levels and the total local fighting intensity.
Lemma 3. Both the local hostility levels Γβ,γi (G) and the total local fighting intensity Λβ,γ(G) are
decreasing with β, and increasing with γ. Moreover if γ < 1/d−max then Γβ,γi (G) ≥ 0, and we have
that 0 ≤ Λβ,0(G) ≤ Λβ,γ(G) ≤ Λ0,γ(G) ≤ 1.
Proof of Lemma 3. We first analyze changes in the local hostility levels with chaining the pa-rameters β and γ. Note that
∂Γβ,γi (G)
∂β= −
d+i(1 + βd+i − γdi−)2
= −d+i Γβ,γi (G)2 ≤ 0,
and∂Γβ,γ
i (G)
∂γ=
d−i(1 + βd+i − γd−i )
2= d−i Γ
β,γi (G)2 ≥ 0,
Similarly, we find that
∂Λβ,γ(G)
∂β= −
∑ni=1 d
+i Γ
β,γi (G)2
(∑ni=1 Γ
β,γi (G)
)2 ≤ 0,
and∂Λβ,γ(G)
∂γ=
∑ni=1 d
−i Γ
β,γi (G)2
(∑ni=1 Γ
β,γi (G)
)2 ≥ 0.
Hence, both Γβ,γi (G) and Λβ,γ(G) are decreasing with β, and increasing with γ. Hence, it follows
thatΛβ,0(G) ≤ Λβ,γ(G) ≤ Λ0,γ(G).
We further have that
Γβ,0i (G) →
{1, as β → 0,
11+d+i
, as β → 1,
and
Λβ,0(G) →
{1− 1
n , as β → 0,
1− 1nH (H+), as β → 1,
where H+ is the graph obtained by adding a loop to each node in G+, that is, the adjacency matrixof H+ is given by In +A+, and
H (H+) ≡1
1n
∑ni=1
1di(H+)
=1
1n
∑ni=1
1d+i +1
is the harmonic mean of the degrees in H+. The maximum degree of H+ is bounded by n (n − 1
in G+), the harmonic mean H (H+) is bounded by n, and it follows from ∂Λn,β
∂β ≤ 0 that
0 ≤ 1−β
nH (H+) ≤ Λβ,0(G) ≤ 1−
1
n≤ 1.
1
In the case of β = 1 (perfect substitutes), Λ1,0(G) = 1 − 1nH (H+) is an inverse measure of the
network density. In a complete graph Kn, we obtain Λ1,0(Kn) = 0, while in an empty graph Kn,we get Λ1,0(Kn) = 1. It follows that in the case of γ = 0 and perfect substitutes when β = 1,aggregate operational performance vanishes in the complete network, and is highest in the emptynetwork, i.e. the standard Tullock contest success game without effort spillovers.
Further, requiring that γ < 1/d−max it follows that
Γβ,γi =
1
1 + βd+i − γd−i≥ Γ0,γ
i =1
1− γd−i> 0,
and the local hostility levels are non-negative for all i = 1, . . . , n. It then follows that
1− Λ0,γ(G) =1
∑ni=1 Γ
0,γi
≥ 0,
which is equivalent to Λ0,γ(G) ≤ 1. Hence, we have that 0 ≤ Λβ,0(G) ≤ Λβ,γ(G) ≤ Λ0,γ(G) ≤ 1.
We next provide a complete characterization for the key player strategy. The following propo-sition characterizes the key player in terms of his position in the network.
Proposition 2. Let G\{i} be the network obtained from G by removing agent i and assume thatthe conditions in Proposition 1 hold. Then the key player i∗ ∈ N ≡ {1, . . . , n} ∪ ∅ is given by
i∗ = argmaxi∈N
{RDβ,γ(G)− RDβ,γ(G\{i})
},
where
RDβ,γ(G)− RDβ,γ(G\{i})
= V Λβ,γ(G)(1 − Λβ,γ(G))
n∑
j=i
cβ,γj (G) +∑
j 6=i
hβ,γi (G)(1 − (1− Λβ,γ(G))hβ,γi (G))
1− Λβ,γ(G)
×n∑
k=1
[(mβ,γ
jk (G)−mβ,γ
ij (G)mβ,γik (G)
mβ,γii (G)
)Γβ,γk (G)
(1 + βd+k − γd−k
1 + β(d+k − 1)− γd−k1{k∈N+
i }
+1 + βd+k − γd−k
1 + βd+k − γ(d−k − 1)1{k∈N−
i } + 1{k/∈(N+i ∪N−
i )}
)]}, (41)
and we have defined by
hβ,γi (G) ≡
1− β
∑
j∈N+i
Γβ,γj (G)(1 − Λβ,γ(G))
1 + β(d+j − 1)− γd−j− γ
∑
j∈N−i
Γβ,γj (G)(1 − Λβ,γ(G))
1 + βd+j − γ(d−j − 1)
−1
,
with mβ,γij (G) being the ij-th element of the matrix Mβ,γ(G) = (In + βA+ − γA−)−1.
Proof of Proposition 2. Let G\{i} be the network obtained from G by removing agent i. Thenthe key player i∗ ∈ N = {1, . . . , n} ∪ ∅ is given by
i∗ = argmaxi∈N
{RDβ,γ(G)− RDβ,γ(G\{i})
}.
2
We can write the change in rent dissipation due to the removal of agent i as follows
RDβ,γ(G) − RDβ,γ(G\{i}) = V Λβ,γ(G)(1 − Λβ,γ(G))
×
n∑
j=1
cβ,γj (G)−Λβ,γ(G\{i})(1 − Λβ,γ(G\{i}))
Λβ,γ(G)(1 − Λβ,γ(G))
n∑
j=1
cβ,γj (G\{i})
.
With∑n
j=1 cβ,γj (G) =
∑nj=1m
β,γjk (G)Γβ,γ
k (G), where mβ,γij (G) is the ij-th element of the matrix
Mβ,γ(G) = (In + βA+ − γA−)−1, we can write this as
RDβ,γ(G) − RDβ,γ(G\{i}) = V Λβ,γ(G)(1 − Λβ,γ(G))
cβ,γi (G) +
∑
j 6=i
∑
k
(mβ,γ
jk (G)Γβ,γk (G)
−Λβ,γ(G\{i})(1 − Λβ,γ(G\{i}))
Λβ,γ(G)(1 − Λβ,γ(G))mβ,γ
jk (G\{i})Γβ,γk (G\{i})
)].
Using the fact that (cf. Lemma 1 in Ballester, Calvo-Armengol and Zenou 2006)
mβ,γjk (G\{i}) = mβ,γ
jk (G) −mβ,γ
ij (G)mβ,γik (G)
mβ,γii (G)
,
and denoting by
hβ,γi (G) ≡Λβ,γ(G\{i})(1 − Λβ,γ(G\{i}))
Λβ,γ(G)(1 − Λβ,γ(G))
=
1− β
∑
j∈N+i
Γβ,γj (1− Λβ,γ(G))
1 + β(d+j − 1)− γd−j− γ
∑
j∈N−i
Γβ,γj (1− Λβ,γ(G))
1 + βd+j − γ(d−j − 1)
−1
,
one can show that
RDβ,γ(G) − RDβ,γ(G\{i}) = V Λβ,γ(G)(1 − Λβ,γ(G))
n∑
j=1
cβ,γj (G) +∑
j 6=i
hβ,γi (G)(1 − (1− Λβ,γ(G)))hβ,γi (G)
1− Λβ,γ(G)
×n∑
k=1
[(mβ,γ
jk (G) −mβ,γ
ij (G)mβ,γik (G)
mβ,γii (G)
)Γβ,γk (G)
(1 + βd+k − γd−k
1 + β(d+k − 1)− γd−k1{k∈N+
i }
+1 + βd+k − γd−k
1 + βd+k − γ(d−k − 1)1{k∈N−
i } + 1{k/∈(N+i ∪N−
i )}
)]}, (42)
where 1{k∈N+i } and 1{k∈N−
i } are indicator variables taking the value of one if, respectively, k ∈
N+i and k ∈ N−
i , and zero otherwise. Then, the key player can be computed explicitly as
i∗ = argmaxi∈N
n∑
j=1
cβ,γj (G) +∑
j 6=i
hβ,γi (G) (1−(1− Λβ,γ (G)
)hβ,γi (G))
1− Λβ,γ (G)
×n∑
k=1
[(mβ,γ
jk (G)−mβ,γ
ij (G)mβ,γik (G)
mβ,γii (G)
)Γβ,γk (G)
(1 + βd+k − γd−k
1 + β(d+k − 1
)− γd−k
1{k∈N+i }
+1 + βd+k − γd−k
1 + βd+k − γ(d−k − 1
)1{k∈N−i } + 1{k/∈(N+
i ∪N−i )}
)]}. (43)
3
Remark 2. The key player identified in Proposition 2 differs from the one introduced in Ballester,Calvo-Armengol and Zenou (2006). In the latter, the key player is defined as i∗ = argmaxi∈Nbu,i(G,α)/Wi(G,α) with bu,i(G,α) being the Bonacich centrality of agent i in G and Wi(G,α)being the generating function of the number of closed walks that start and terminate at node i.47
Compared with our key player formula in equation (41) we find that it is more involved. This isnot surprising as the contest success function makes our payoff function generically non-linear.
In the following we provide a complete equilibrium characterization of the extension we haveintroduced in Section 2.6. First, let assume that the fighting strength ϕi of agent i depends on anidiosyncratic shifter ϕi as in equation (13). Then the following proposition characterizes the localNash equilibrium.
Proposition 3. Assume that β + γ < 1/max{λmax(G+), d−max}, where λmax (A
±) denotes the
largest eigenvalue associated with the matrix A±. Let Γβ,γi (G) and Λβ,γ (G) be defined as in equation
(6), and let
cβ,γµ
(G) ≡(In + βA+ − γA−
)−1µ (44)
be a centrality vector, whose generic element cβ,γµ,i (G) describes the centrality of agent i in the
network for some vector µ ∈ Rn. Then there exists a unique local Nash equilibrium of the n–
player simultaneous move game with payoffs given by equation (1), agents’ OPs in equation (2)and strategy space S = [x1,∞) × [x2,∞) × . . . × [xn,∞) ⊂ R
n, where the equilibrium effort levelsare given by
x∗i (G) = V Λβ,γ (G)(1− Λβ,γ (G)
)cβ,γΓβ,γ(G),i
(G)− cβ,γϕ,i(G), (45)
for all i = 1, . . . , n. Moreover, the equilibrium OPs are given by equation (26), and the equilibriumpayoffs are given by
π∗i (G) ≡ πi (x∗, G) = V (1− Λβ,γ(G))
(Γβ,γi (G)− Λβ,γ (G) cβ,γ
Γβ,γ(G),i(G))+ cβ,γ
ϕ,i (G). (46)
Proof of Proposition 3. In order to compute the equilibrium we start with deriving the necessaryfirst order conditions (FOCs). With equation (13) we can write the payoff of agent i as follows
πi(G,x) = Vϕi∑nj=1 ϕj
− xi
= Vxi + β
∑nj=1 a
+ijxj − γ
∑nj=1 a
−ijxj + ϕi
∑nj=1
(xj + β
∑nk=1 a
+jkxk − γ
∑nk=1 a
−jkxk + ϕj
) − xi. (47)
The partial derivatives are given by
∂πi∂xi
= V
∂ϕi∂xi
∑nj=1 ϕj − ϕi
∑nj=1
∂ϕj
∂xi(∑nj=1 ϕj
)2 − 1
= V
∑nj=1ϕj − ϕi(1 + βd+i − γd−i )(∑n
j=1 ϕj
)2 − 1, (48)
47See also equation (61) in the Online Appendix D.
4
where we have used the fact that∂ϕj
∂xi= δij+βa
+ij−γa
−ij and consequently
∑nj=1
∂ϕj
∂xi= 1+βd+i −γd
−i .
The first order conditions are then given by ∂πi∂xi
= 0. From the partial derivative in equation (48)the FOC can be written as follows
∂πi∂xi
= V
∑nj=1 ϕj − ϕi(1 + βd+i − γd−i )(∑n
j=1ϕj
)2 − 1 = 0,
from which we get
ϕi =1
1 + βd+i − γd−i
1−
1
V
n∑
j=1
ϕj
n∑
j=1
ϕj .
Summation over i givesn∑
i=1
ϕi = V
1−
1∑ni=1
11+βd+i −γd−i
.
With Γβ,γi (G) and Λβ,γ(G) as in equation (6) we can write the aggregate operational performance
asn∑
i=1
ϕi = V Λβ,γ(G),
which is equivalent to equation (26). The individual operational performance can be written as
ϕi(G,x) = V Λβ,γ(G)(1 − Λβ,γ(G))Γβ,γi (G), (49)
which is equivalent to equation (26). We then get
ϕi(G,x) = xi + β
n∑
j=1
a+ijxj − γ
n∑
j=1
a−ijxj + ϕi = V Λn,β(G)(1 − Λn,β(G))Γβ,γi (G). (50)
We can write
xi + βn∑
j=1
a+ijxj − γn∑
j=1
a−ijxj = ϕi − ϕi = V Λn,β(G)(1 − Λn,β(G))Γβ,γi (G) − ϕi. (51)
Denoting by Γβ,γ(G) ≡ (Γβ,γ1 (G), . . . ,Γβ,γ
n (G))⊤, we can write this in vector-matrix form as
(In + βA+ − γA−)x = V Λβ,γ(G)(1 − Λβ,γ(G))Γβ,γ(G)− ϕ.
When the matrix In + βA+ − γA− is invertible (see the proof of Proposition 1 for the conditionsthat guarantee invertibility), we obtain a unique solution given by
x = V Λβ,γ(G)(1 − Λβ,γ(G))(In + βA+ − γA−)−1Γβ,γ(G)− (In + βA+ − γA−)−1ϕ. (52)
With the definition of the centrality in equation (44) we then can write equation (52) in the formof equation (45) in the proposition. Moreover, using the fact that equilibrium payoffs are given by
πi(x∗, G) = V
ϕ∗i (G)∑n
j=1 ϕ∗j (G) − x∗i we obtain equation (46) in the proposition.
Further, the second order partial derivatives, ∂2πi
∂x2i, are the same as in the proof of Proposition
1. In particular, it follows that ∂2πi
∂x2i< 0 if Λβ,γ(G) > 0 and 1+βd+i − γd−i > 0. The last inequality
5
requires that the local hostility level of agent i is positive, i.e. Γβ,γi (G) > 0. It then follows that
the equilibrium payoff function is locally concave at the efforts levels in equation (52).As in the proof of Proposition 1 in the remainder of the proof we show that the equilibrium
payoff function is globally concave, by bounding the effort levels (from below). This implies that themarginal payoff of an agent from deviating from the Nash equilibrium strategy is always negative.Observe that
πi(x∗−i, xi, G) = V
ϕi(x∗−i, x,G)∑n
j=1 ϕj(x∗−i, x,G)
− xi
= Vϕi(x
∗, G)− (x∗i − xi)∑nj=1 ϕj(x
∗, G) − (1 + βd+i − γd−i )(x∗i − xi)
− xi
= VAβ,γ
i (G) + Γβ,γi (G)xi
Bβ,γi (G) + xi
− xi, (53)
where we have denoted by
Aβ,γi (G) ≡ Γβ,γ
i (G)(ϕ∗i − x∗i ) = V Λβ,γ(G)(1 − Λβ,γ(G))Γβ,γ
i (G)(Γβ,γi (G)− cβ,γi (G)
)+ Γβ,γ
i (G)cβ,γi (G),
Bβ,γi (G) ≡ Γβ,γ
i (G)
n∑
j=1
ϕ∗j − x∗i = V Λβ,γ(G)Γβ,γ
i (G)− x∗i
= V Λβ,γ(G)(Γβ,γi (G)− (1− Λβ,γ(G))cβ,γi (G)
)+ cβ,γi (G),
the centrality cβ,γi (G) has been defined in equation (7) and we have denoted by cβ,γ(G) ≡ (In +
βA+ − γA−)−1ϕ. Assume first that Bβ,γ
i (G) ≥ 0. For xi > −Bβ,γi (G) the function πi(x
∗−i, x,G) is
concave in xi with a maximum at x∗i , and decreasing for values of xi > x∗i with an asymptote given
by −xi.48 In particular, if xi is larger than the discontinuity at −Bβ,γ
i (G) of πi(x∗−i, x,G) then we
have that πi(x∗−i, xi, G) < πi(x
∗, G). In particular, we have that πi(x∗−i, xi, G) < πi(x
∗, G) iff
VAβ,γ
i (G) + Γβ,γi (G)xi
Bβ,γi (G) + xi
− xi < π∗i = Vϕ∗i∑n
j=1ϕ∗j(G)
− x∗i
= V (1− Λβ,γ(G))Γβ,γi (G) − x∗i
= V (1− Λβ,γ(G))(Γβ,γi (G)− Λβ,γ(G)cβ,γi (G)
)+ cβ,γi (G), (54)
where we have denoted by π∗i ≡ πi(x∗, G). Next, observe that from the FOC
∂πi(x∗−i, xi, G)
∂xi= −
Aβ,γi (G)−Bβ,γ
i (G)Γβ,γi (G) + (Bβ,γ
i (G) + xi)2
(Bβ,γi (G) + xi)2
= 0, (55)
it follows that49
xi = −Bβ,γi (G) +
√Bβ,γ
i (G)Γβ,γi (G) −Aβ,γ
i (G) = x∗i ,
48See footnote 45.49W.l.o.g. assume that V = 1, Then using the fact that Aβ,γ
i (G) = Γβ,γi (G)Λβ,γ(G)(1 − Λβ,γ(G))(Γβ,γ
i (G) −cβ,γi (G)) + Γβ,γ
i (G)cβ,γi (G), Bβ,γ
i (G) = Λβ,γ(G)(Γβ,γi (G) − (1 − Λβ,γ(G))cβ,γ
i (G)) + cβ,γi (G) and x∗
i = Λβ,γ(G)(1 −
Λβ,γ(G))cβ,γi (G)− cβ,γi (G), we find that −Bβ,γ
i (G)+√
Bβ,γi (G)Γβ,γ
i (G)− Aβ,γi (G) = Λβ,γ(G)(1−Λβ,γ(G))cβ,γi (G)−
cβ,γi (G) = x∗
i .
6
and inserting into πi(x∗−i, xi, G) yields π∗i . We then define xi ≡ −Bβ,γ
i (G). When Bβ,γi (G) ≥ 0 it
then follows that xi < 0. Note that if Bβ,γi (G) > 0 then the sign of Aβ,γ
i (G) does not qualitativelychange the functional form of πi(x
∗−i, x,G) (more precisely, πi(x
∗−i, x,G) is a concave function for
xi ≥ −Bβ,γi (G) with a global maximum over its domain at x∗i ; see also footnote 45). In contrast,
if Bβ,γi (G) < 0 then we must have that Aβ,γ
i (G) must be negative as well. More precisely, if
Bβ,γi (G) < 0 then we must have that Aβ,γ
i (G) < Bβ,γi (G)Γβ,γ
i (G). Otherwise we would have
that∂πi(x∗
−i,xi,G)
∂xi< 0 (see equation (55)), violating the FOC. However, under this condition, the
functional form of πi(x∗−i, x,G) remains qualitatively unchanged. Importantly, for all the above
cases we find that πi(x∗−i, xi, G) − πi(x
∗, G) < 0 if xi ∈ [xi,∞), and we obtain a local Nashequilibrium (see Definition 2). This completes the proof.
Remark 3. Similar to Remark 1 it is possible to compute a common lower bound, x, on theeffort levels in Proposition 3 such that πi(x
∗−i, xi, G) − πi(x
∗, G) < 0 holds for all i = 1, . . . , nif x ∈ [x,∞)n. We also require that x < x∗i for all i = 1, . . . , n. These conditions then implya symmetric local Nash equilibrium (see Definition 2). We denote this common lower bound asx ≡ maxi=1,...,n xi, and we require that ∀i = 1, . . . , n
x ≤ x∗i = V Λβ,γ(G)(1 − Λβ,γ(G))cβ,γi (G) − cβ,γi (G). (56)
Observe that equation (56) is equivalent to
maxi=1,...,n
{−Bβ,γ
i (G)}= max
i=1,...,n
x
∗i − Γβ,γ
i (G)n∑
j=1
ϕ∗j
≤ min
i=1,...,nx∗i ,
which ismax
i=1,...,n
{x∗i − V Λβ,γ(G)Γβ,γ
i (G)}≤ min
i=1,...,nx∗i ,
or equivalently
maxi=1,...,n
{V Λβ,γ(G)
((1− Λβ,γ(G))cβ,γi (G)− Γβ,γ
i (G))− cβ,γi (G)
}
≤ mini=1,...,n
{V Λβ,γ(G)(1 − Λβ,γ(G))cβ,γi (G) − cβ,γi (G)
}. (57)
7
Online Appendix C Tables for Key Player Analysis and Pacifica-
tion Policies
In this appendix we provide additional tables complementing the discussion in Sections 5.2 and 5.3.
8
Appendix Table I: Key player ranking for the first 35 actors in the Democratic Republic of the Congo (DRC) excluding government groups.a
The total number of groups is n = 85. The total number of alliances is m+(Gb) = 95, the total number of conflicts is m−(Gb) = 111 and wehave that
∑j ϕ
∗j(G
b) = 0.9891. The Pearson correlation of fighting effort in the benchmark with the reduction in the rent dissipation is 0.76.
Actor ∆RDβ,γk (%)b ϕ∗
i (Gb)/
∑
j ϕ∗j (G
b) (%) x∗i (G
b)/RDβ,γ(Gb) (%) Rank x∗i (G
b) Rank ∆RDβ,γk
FDLR: Dem. Forces for the Lib. of Rwanda 13.79 0.71 6.65 4 1Mayi-Mayi Milita 9.67 0.69 5.59 6 2Mil. Forc. of Rwanda (2000-) 9.57 1.91 3.65 10 3Mil. Forc. of Uganda (1986-) 7.71 0.93 4.34 8 4Lendu Ethnic Militia (DRC) 4.97 1.24 2.43 13 5Mil. Forc. of Rwanda (1994-1999) 4.56 1.18 1.69 16 6RCD: Rally for Congolese Dem. 4.49 0.98 1.75 15 7Interahamwe Hutu Ethnic Militia 2.36 0.84 1.39 18 8RCD: Rally for Congolese Dem. (Goma) 2.36 6.90 8.67 3 9UPC: Union of Congolese Patriots 1.74 1.58 2.16 14 10Mil. Forc. of Zimbabwe (1980-) 1.72 0.67 0.73 24 11Hutu Rebels 1.46 0.99 0.52 27 12Mil. Forc. of South Africa (1994-1999) 0.80 0.91 0.28 35 13FAA/MPLA: Mil. Forc. of Angola (1975-) 0.75 0.76 0.36 31 14Mayi Mayi Militia (PARECO) 0.72 0.68 0.37 29 15Mil. Forc. of Namibia (1990-2005) 0.62 0.76 0.31 32 16PUSIC: Party Unity & Safe. of Congo’s Integrity 0.54 1.33 0.26 36 17Former Mil. Forc. of Rwanda (1973-1994) 0.45 0.79 0.24 37 18Mil. Forc. of Sudan (1993-) 0.41 0.95 0.24 38 19Mil. Forc. of Burundi (1996-2005) 0.27 1.20 0.29 33 20Mayi-Mayi Militia (Yakutumba) 0.25 0.70 0.15 42 21PPRD: People’s Party for Reconstr. & Dem. 0.13 1.09 0.13 43 22ADFL: All. of Dem. Forces for Lib. (Congo-Zaire) (1996-1997) 0.11 1.33 0.06 55 23Munzaya Ethnic Militia (DRC) 0.11 1.09 0.11 45 24Mil. Forc. of Zambia (1991-2002 ) 0.10 1.20 0.05 62 25Mayi Mayi Milita (Cmdt La Fontaine) 0.10 0.82 0.06 56 26Mayi-Mayi Militia (Cmdt Jackson) 0.10 0.82 0.06 57 27Wageregere Ethnic Militia 0.10 1.20 0.05 61 28Mil. Forc. of Chad (1990-) 0.08 0.82 0.05 63 29RCD: Rally for Congolese Dem. (Nat.) 0.06 0.70 0.03 72 30Ngiti Ethnic Militia 0.06 0.96 0.03 70 31Bomboma Ethnic Militia 0.06 1.20 0.08 50 32RUD: Gathering for Unity & Dem. 0.06 1.20 0.05 60 33Mayi-Mayi Militia (Kifuafua) 0.05 0.77 0.03 73 34Alur Ethnic Militia (Uganda) 0.05 0.96 0.03 71 35
a We have used the parameter estimates β = 0.1407 and γ = 0.0903 while setting V = 1.b The relative change in the rent dissipation is computed as ∆RDβ,γ
k (%) =(
RDβ,γ(Gb)− RDβ,γ(Gb\{k}))
/RDβ,γ(Gb), while the rent dissipation in the original
network Gb is RDβ,γ(Gb) =∑n
i=1 x∗i (G
b) = 474.4614 (with V = 1). A positive value of ∆RDβ,γk thus indicates a reduction in fighting, while a negative value indicates
an increase in fighting.
9
(continued) Appendix Table I: Key player ranking for the 36-th to the 80-th actors in the Democratic Republic of the Congo (DRC) excludinggovernment groups.
Actor ∆RDβ,γk (%) ϕ∗
i (Gb)/
∑
j ϕ∗j (G
b) (%) x∗i (G
b)/RDβ,γ(Gb) (%) Rank x∗i (G
b) Rank ∆RDβ,γk
Lendu Ethnic Militia (Uganda) 0.04 1.20 0.06 54 36FLC: Congolese Lib. Front 0.04 1.49 0.16 40 37Group of 47 0.03 1.09 0.03 68 38Banyamulenge Ethnic Militia (DRC) 0.03 1.09 0.03 67 39Mbingi Community Militia (DRC) 0.03 1.09 0.03 69 40Mayi Mayi Militia (Mbuayi) 0.03 0.77 0.02 83 41All. for Dem. Change 0.02 0.96 0.02 82 42DSP: Division Speciale Presidentelle 0.02 0.96 0.02 81 43FAP: Popular Self-Defense Forces 0.01 1.09 0.02 74 44Hutu Refugees (Rwanda) 0.01 1.09 0.02 77 45Mutiny of LRA: Lord’s Resistance Army 0.01 1.09 0.02 75 46PRA: People’s Redemption Army 0.01 1.09 0.02 76 47Mil. Forc. of Zaire (1965-1997) 0.01 1.09 0.02 78 48Wangilima Ethnic Militia 0.01 1.09 0.02 79 49Pygmy Ethnic Group (DRC) 0.01 1.09 0.02 80 50Faustin Munene Militia (DRC) 0.00 1.09 0.00 84 51UNAFEC: Union of Nat. Fed. of Congo Party (DRC) 0.00 1.09 0.00 85 52CNDD-FDD (Ndayikengurukiye faction) -0.03 1.04 0.03 66 53RCD: Rally for Congolese Dem. (Masunzu) -0.04 0.82 0.05 59 54UPPS: Union for Dem. & Social Progr. Party (DRC) -0.04 1.33 0.03 65 55FRF: Federal Republican Forces -0.08 1.20 0.05 58 56Haut-Uele Resident Militia -0.09 1.04 0.06 53 57Minembwe Dissidents -0.10 1.20 0.06 52 58APCLS: All. of Patriots for a Free & Sov. Congo -0.11 1.04 0.06 51 59Unnamed Mayi-Mayi Militia (DRC) -0.14 1.04 0.08 49 60Lobala (Enyele) Militia -0.14 1.33 0.10 46 61FNL: Nat. Forces of Lib. -0.16 1.14 0.10 47 62Enyele Ethnic Militia (DRC) -0.17 1.20 0.11 44 63UNITA: Nat. Union for the Total Indep. of Angola -0.27 0.99 0.15 41 64BDK: Bunda Dia Kongo -0.29 1.20 0.19 39 65CNDD-FDD: Nat. Council for the Defence of Dem. -0.37 1.71 0.71 25 66ALIR: Army for the Lib. of Rwanda -0.42 1.33 0.36 30 67NALU: Nat. Army for the Lib. of Uganda -0.56 0.99 0.28 34 68MRC: Rev. Movement of Congo -0.56 0.91 0.47 28 69SPLA/M: Sudanese People’s Lib. Army/Movement -0.87 1.04 0.73 23 70FNI: Nat. & Integr. Front -1.17 1.04 0.84 22 71FPJC: Popular Front for Justice in Congo -1.32 1.33 0.65 26 72FRPI: Front for Patr. Resist. of Ituri -1.34 1.14 0.86 21 73Hema Ethnic Militia (DRC) -1.78 0.98 0.94 20 74ADF: Allied Dem. Forces -2.40 1.08 1.33 19 75Mutiny of Mil. Forc. of DRC (2003-) -3.30 1.08 1.64 17 76MLC: Congolese Lib. Movement -5.22 1.07 3.08 11 77RCD: Rally for Congolese Dem. (Kisangani) -5.46 4.39 5.95 5 78LRA: Lord’s Resistance Army -7.54 2.96 4.13 9 79CNDP: Nat. Congress for the Defense of the People -8.78 1.18 4.35 7 80
10
Appendix Table II: Key links ranking for the first 35 actors in the Democratic Republic of the Congo (DRC) excluding government groups.a
The total number of groups is n = 85. The total number of alliances is m+(Gb) = 95, the total number of conflicts is m−(Gb) = 111 and wehave that
∑j ϕ
∗j(G
b) = 0.9891. The Pearson correlation of fighting effort in the benchmark with the reduction in the rent dissipation is 0.72.
Actor ∆RDβ,γkj (%)b ϕ∗
i (Gb)/
∑
j ϕ∗j (G
b) (%) x∗i (G
b)/RDβ,γ(Gb) (%) Rank x∗i (G
b) Rank ∆RDβ,γkj
RCD: Rally for Congolese Dem. (Goma) 17.21 6.90 8.67 3 1FDLR: Dem. Forces for the Lib. of Rwanda 9.68 0.71 6.65 4 2RCD: Rally for Congolese Dem. (Kisangani) 8.74 4.39 5.95 5 3LRA: Lord’s Resistance Army 4.30 2.96 4.13 9 4CNDP: Nat. Congress for the Defense of the People 2.53 1.18 4.35 7 5Lendu Ethnic Militia (DRC) 2.47 1.24 2.43 13 6Mil. Forc. of Rwanda (2000-) 0.60 1.91 3.65 10 7Mil. Forc. of Zimbabwe (1980-) 0.29 0.67 0.73 24 8Mayi-Mayi Militia (Cmdt Jackson) 0.27 0.82 0.06 57 9Mayi Mayi Milita (Cmdt La Fontaine) 0.27 0.82 0.06 56 10Mil. Forc. of Uganda (1986-) 0.27 0.93 4.34 8 11Mayi-Mayi Milita 0.25 0.69 5.59 6 12Mil. Forc. of Sudan (1993-) 0.25 0.95 0.24 38 13Mayi Mayi Militia (Mbuayi) 0.24 0.77 0.02 83 14Mayi-Mayi Militia (Kifuafua) 0.24 0.77 0.03 73 15Mil. Forc. of Chad (1990-) 0.24 0.82 0.05 63 16Mayi-Mayi Militia (Yakutumba) 0.24 0.70 0.15 42 17Former Mil. Forc. of Rwanda (1973-1994) 0.23 0.79 0.24 37 18Mayi Mayi Militia (PARECO) 0.23 0.68 0.37 29 19FAA/MPLA: Mil. Forc. of Angola (1975-) 0.22 0.76 0.36 31 20Mil. Forc. of Namibia (1990-2005) 0.22 0.76 0.31 32 21Interahamwe Hutu Ethnic Militia 0.20 0.84 1.39 18 22MLC: Congolese Lib. Movement -1.98 1.07 3.08 11 23Mil. Forc. of Rwanda (1994-1999) -3.08 1.18 1.69 16 24UPC: Union of Congolese Patriots -3.27 1.58 2.16 14 25RCD: Rally for Congolese Dem. -3.43 0.98 1.75 15 26Hutu Rebels -5.28 0.99 0.52 27 27Mil. Forc. of South Africa (1994-1999) -5.68 0.91 0.28 35 28PUSIC: Party Unity & Safe. of Congo’s Integrity -7.20 1.33 0.26 36 29Mil. Forc. of Zambia (1991-2002 ) -7.56 1.20 0.05 62 30Wageregere Ethnic Militia -7.75 1.20 0.05 61 31RCD: Rally for Congolese Dem. (Nat.) -7.76 0.70 0.03 72 32Alur Ethnic Militia (Uganda) -8.01 0.96 0.03 71 33All. for Dem. Change -8.07 0.96 0.02 82 34Ngiti Ethnic Militia -8.14 0.96 0.03 70 35
a We have used the parameter estimates β = 0.1407 and γ = 0.0903 while setting V = 1.b The relative change in the rent dissipation is computed as ∆RDβ,γ
kj (%) =(
RDβ,γ(Gb)− RDβ,γ(Gb\{kj}))
/RDβ,γ(Gb), while the rent dissipation in the original
network Gb is RDβ,γ(Gb) =∑n
i=1 x∗i (G
b) = 474.4614 (with V = 1). A positive value of ∆RDβ,γkj thus indicates a reduction in fighting, while a negative value indicates
an increase in fighting.
11
(continued) Appendix Table II: Key links ranking for the 36-th to the 80-th actors in the Democratic Republic of the Congo (DRC) excludinggovernment groups.
Actor ∆RDβ,γkj (%) ϕ∗
i (Gb)/
∑
j ϕ∗j (G
b) (%) x∗i (G
b)/RDβ,γ(Gb) (%) Rank x∗i (G
b) Rank ∆RDβ,γkj
DSP: Division Speciale Presidentelle -8.19 0.96 0.02 81 36Mil. Forc. of Burundi (1996-2005) -8.20 1.20 0.29 33 37ADFL: All. of Dem. Forces for Lib. (Congo-Zaire) (1996-1997) -8.23 1.33 0.06 55 38PPRD: People’s Party for Reconstr. & Dem. -8.49 1.09 0.13 43 39RUD: Gathering for Unity & Dem. -8.54 1.20 0.05 60 40Munzaya Ethnic Militia (DRC) -8.54 1.09 0.11 45 41CNDD-FDD: Nat. Council for the Defence of Dem. -8.60 1.71 0.71 25 42Banyamulenge Ethnic Militia (DRC) -8.80 1.09 0.03 67 43Group of 47 -8.80 1.09 0.03 68 44Mbingi Community Militia (DRC) -8.80 1.09 0.03 69 45FAP: Popular Self-Defense Forces -8.85 1.09 0.02 74 46PRA: People’s Redemption Army -8.85 1.09 0.02 76 47Mutiny of LRA: Lord’s Resistance Army -8.85 1.09 0.02 75 48Hutu Refugees (Rwanda) -8.85 1.09 0.02 77 49Mil. Forc. of Zaire (1965-1997) -8.85 1.09 0.02 78 50Wangilima Ethnic Militia -8.85 1.09 0.02 79 51Pygmy Ethnic Group (DRC) -8.85 1.09 0.02 80 52Faustin Munene Militia (DRC) -8.90 1.09 -0.00 84 53UNAFEC: Union of Nat. Fed. of Congo Party (DRC) -8.90 1.09 -0.00 85 54Bomboma Ethnic Militia -9.05 1.20 0.08 50 55FLC: Congolese Lib. Front -9.18 1.49 0.16 40 56Lendu Ethnic Militia (Uganda) -9.28 1.20 0.06 54 57SPLA/M: Sudanese People’s Lib. Army/Movement -9.84 1.04 0.73 23 58FNI: Nat. & Integr. Front -10.15 1.04 0.84 22 59MRC: Rev. Movement of Congo -10.34 0.91 0.47 28 60RCD: Rally for Congolese Dem. (Masunzu) -10.90 0.82 0.05 59 61Unnamed Mayi-Mayi Militia (DRC) -10.97 1.04 0.08 49 62ALIR: Army for the Lib. of Rwanda -10.99 1.33 0.36 30 63FRPI: Front for Patr. Resist. of Ituri -11.16 1.14 0.86 21 64FNL: Nat. Forces of Lib. -11.44 1.14 0.10 47 65Hema Ethnic Militia (DRC) -11.52 0.98 0.94 20 66Mutiny of Mil. Forc. of DRC (2003-) -11.90 1.08 1.64 17 67CNDD-FDD (Ndayikengurukiye faction) -12.13 1.04 0.03 66 68NALU: Nat. Army for the Lib. of Uganda -12.72 0.99 0.28 34 69Haut-Uele Resident Militia -13.06 1.04 0.06 53 70UNITA: Nat. Union for the Total Indep. of Angola -13.48 0.99 0.15 41 71BDK: Bunda Dia Kongo -13.86 1.20 0.19 39 72Enyele Ethnic Militia (DRC) -14.22 1.20 0.11 44 73UPPS: Union for Dem. & Social Progr. Party (DRC) -14.26 1.33 0.03 65 74Minembwe Dissidents -14.44 1.20 0.06 52 75FRF: Federal Republican Forces -14.51 1.20 0.05 58 76Lobala (Enyele) Militia -14.53 1.33 0.10 46 77APCLS: All. of Patriots for a Free & Sov. Congo -14.73 1.04 0.06 51 78FPJC: Popular Front for Justice in Congo -16.73 1.33 0.65 26 79ADF: Allied Dem. Forces -37.28 1.08 1.33 19 80
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Online Appendix D Bonacich Centrality
In this section we discuss a network measure capturing the centrality of an agent in the networkdue to Bonacich (1987). Let A be the symmetric n × n adjacency matrix of the network G andλmax its largest real eigenvalue. The matrix M(G,α) = (I−αA)−1 exists and is non-negative ifand only if α < 1/λmax.
50 Under this condition it can be written as
M(G,α) =∞∑
k=0
αkAk. (58)
The vector of Bonacich centralities is then given by
bu(G,α) = M(G,α)u, (59)
where u = (1, . . . , 1)⊤ is an n-dimensional vector of ones. We can write the vector of Bonacichcentralities as
bu(G,α) = (I− αA)−1u =
∞∑
k=0
αkAku.
For the components bu,i(G,α), i = 1, . . . , n, we get
bu,i(G,α) =∞∑
k=0
αk(Ak · u)i =∞∑
k=0
αkn∑
j=1
a[k]ij , (60)
where a[k]ij is the ij-th element of Ak. Because Nk,i(G) ≡
∑nj=1 a
[k]ij counts the number of all walks
of length k in G starting from i, the Bonacich centrality of agent i, bu,i(G,α), is thus equivalentto the number of all walks in G starting from i, where the walks of length k are weighted by ageometrically decaying factor αk.
Further, the sum of the Bonacich centralities,∑n
i=1 bu,i(G,α) = u⊤bu(G,α) = u⊤M(G,α)u, isequivalent to walk generating function of the graph G, denoted by N(G,α) (cf. Cvetkovic, 1995).To see this, let Nk(G) ≡
∑ni=1Nk,i(G) denote the number of walks of length k in G. Then we can
write Nk(G) as follows Nk(G) =∑n
i=1
∑nj=1 a
[k]ij = u⊤Aku. The walk generating function is then
defined as
N(G,α) ≡∞∑
k=0
Nk(G)αk = u⊤
(∞∑
k=0
αkAk
)u = u⊤ (In − αA)−1
u = u⊤M(G,α)u.
Moreover, the generating function of the number of closed walks that start and terminate at nodei is given by
Wi(G,α) ≡∞∑
k=0
a[k]ii α
k. (61)
The Bonacich matrix of equation (58) is also a measure of structural similarity of the agentsin the network, called regular equivalence. Blondel et al. (2004) and Leicht et al. (2006) definea similarity score bij, which is high if nodes i and j have neighbors that themselves have highsimilarity, given by bij = α
∑nk=1 aikbkj + δij . In matrix-vector notation this reads M = αAM+ I.
Rearranging yields M = (I − αA)−1 =∑∞
k=0 αkAk, assuming that α < 1/λmax. We hence obtain
50The proof can be found e.g. in Debreu and Herstein (1953).
13
that the similarity matrix M is equivalent to the Bonacich matrix from equation (58). The averagesimilarity of agent i is 1
n
∑nj=1 bij = 1
nbu,i(G,α), where bu,i(G,α) is the Bonacich centrality of i.It follows that the Bonacich centrality of i is proportional to the average regular equivalence of i.Agents with a high Bonacich centrality are then the ones which also have a high average structuralsimilarity with the other agents in the network.
The interpretation of eingenvector-like centrality measures as a similarity index is also importantin the study of correlations between observations in principal component analysis and factor analysis(cf. Rencher and Christensen, 2012). Variables with similar factor loadings can be grouped together.This basic idea has also been used in the economics literature on segregation (e.g. Ballester andVorsatz, 2014; Echenique and Fryer Jr., 2007; Echenique et al., 2006).
References
Blondel, Vincent, Anahi Gajardo, Maureen Heymans, Pierre Senellart, and Paul Van Dooren.2004. ”A measure of similarity between graph vertices: Applications to synonym extraction andweb searching”. Siam Review 46:647–666.
Ballester, Coralio, and Marc Vorsatz. 2014. ”Random walk based segregation measures”, Reviewof Economics and Statistics, forthcoming.
Bonacich, Philip, 1987, ”Power and centrality: A family of measures”, American Journal of Soci-ology 92: 1170.
Debreu, Gerard, and I. N. Herstein. 1953. ”Nonnegative square matrices”, Econometrica 21:597–607.
Echenique, Federico, Fryer Jr., Roland G., 2007, ”A Measure of Segregation Based on Social Inter-actions”, The Quarterly Journal of Economics, 122 (2):441–485.
Echenique, Federico, Fryer Jr., Roland G., Kaufman, Alex, 2006, ”Is school segregation good orbad?”, The American Economic Review, 96(2): 265–269.
Lee, Lung-fei and Xiaodong Liu.2010. ”Efficient GMM estimation of high order spatial autoregres-sive models with autoregressive disturbances”, Econometric Theory, 26:187–230.
Rencher, Alvin C., Christensen, William F., 2012. ”Methods of multivariate analysis”, Vol. 709.John Wiley & Sons.
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